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Guillermo Sapiro

Guillermo Sapiro appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Computer VisionMachine LearningInformation Theorymath.ITmath.OCNeural and Evolutionary ComputingMultimediaApplicationsData Structures and AlgorithmsDiscrete MathematicsDistributed, Parallel, and Cluster Computingeess.IVGenomicsInformation Retrievalmath.COmath.MGmath.SPmath.STNeurons and CognitionNumerical Analysisquant-phQuantitative MethodsSocial and Information NetworksStatistics Theory

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Published work

61 published item(s)

preprint2026arXiv

LLMs are not (consistently) Bayesian: Quantifying internal (in)consistencies of LLMs' probabilistic beliefs

Modern AI systems are being deployed in complex domains such as medicine, science, and law, where it is important that they not only produce correct answers, but also represent and update uncertain beliefs about the world as new evidence arrives. We introduce the novel technique of studying LLMs as information processing rules and utilize the information processing gap to study the internal (in)consistencies of how LLMs update their probabilistic beliefs from evidence. Our extensive experiments evaluate multiple approaches in which LLMs can incorporate evidence into their beliefs. Some of these approaches produce (nearly) Bayesian updates; others seem to use a learned heuristic. Surprisingly, the non-Bayesian heuristic updates often outperform exact Bayesian computation in terms of downstream task performance -- indicating the LLMs' probabilistic models of the world are misspecified. Lastly, we show how our measure can provide diagnostics to identify issues with LLM-powered inferential systems.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Minimax Demographic Group Fairness in Federated Learning

Federated learning is an increasingly popular paradigm that enables a large number of entities to collaboratively learn better models. In this work, we study minimax group fairness in federated learning scenarios where different participating entities may only have access to a subset of the population groups during the training phase. We formally analyze how our proposed group fairness objective differs from existing federated learning fairness criteria that impose similar performance across participants instead of demographic groups. We provide an optimization algorithm -- FedMinMax -- for solving the proposed problem that provably enjoys the performance guarantees of centralized learning algorithms. We experimentally compare the proposed approach against other state-of-the-art methods in terms of group fairness in various federated learning setups, showing that our approach exhibits competitive or superior performance.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Scaling-Translation-Equivariant Networks with Decomposed Convolutional Filters

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST . To reduce the model complexity and computational burden, we decompose the convolutional filters under two pre-fixed separable bases and truncate the expansion to low-frequency components. A further benefit of the truncated filter expansion is the improved deformation robustness of the equivariant representation, a property which is theoretically analyzed and empirically verified. Numerical experiments demonstrate that the proposed scaling-translation-equivariant network with decomposed convolutional filters (ScDCFNet) achieves significantly improved performance in multiscale image classification and better interpretability than regular CNNs at a reduced model size.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

ACDC: Weight Sharing in Atom-Coefficient Decomposed Convolution

Convolutional Neural Networks (CNNs) are known to be significantly over-parametrized, and difficult to interpret, train and adapt. In this paper, we introduce a structural regularization across convolutional kernels in a CNN. In our approach, each convolution kernel is first decomposed as 2D dictionary atoms linearly combined by coefficients. The widely observed correlation and redundancy in a CNN hint a common low-rank structure among the decomposed coefficients, which is here further supported by our empirical observations. We then explicitly regularize CNN kernels by enforcing decomposed coefficients to be shared across sub-structures, while leaving each sub-structure only its own dictionary atoms, a few hundreds of parameters typically, which leads to dramatic model reductions. We explore models with sharing across different sub-structures to cover a wide range of trade-offs between parameter reduction and expressiveness. Our proposed regularized network structures open the door to better interpreting, training and adapting deep models. We validate the flexibility and compatibility of our method by image classification experiments on multiple datasets and underlying network structures, and show that CNNs now maintain performance with dramatic reduction in parameters and computations, e.g., only 5\% parameters are used in a ResNet-18 to achieve comparable performance. Further experiments on few-shot classification show that faster and more robust task adaptation is obtained in comparison with models with standard convolutions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Detecting Adversarial Samples Using Influence Functions and Nearest Neighbors

Deep neural networks (DNNs) are notorious for their vulnerability to adversarial attacks, which are small perturbations added to their input images to mislead their prediction. Detection of adversarial examples is, therefore, a fundamental requirement for robust classification frameworks. In this work, we present a method for detecting such adversarial attacks, which is suitable for any pre-trained neural network classifier. We use influence functions to measure the impact of every training sample on the validation set data. From the influence scores, we find the most supportive training samples for any given validation example. A k-nearest neighbor (k-NN) model fitted on the DNN's activation layers is employed to search for the ranking of these supporting training samples. We observe that these samples are highly correlated with the nearest neighbors of the normal inputs, while this correlation is much weaker for adversarial inputs. We train an adversarial detector using the k-NN ranks and distances and show that it successfully distinguishes adversarial examples, getting state-of-the-art results on six attack methods with three datasets. Code is available at https://github.com/giladcohen/NNIF_adv_defense.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Differential 3D Facial Recognition: Adding 3D to Your State-of-the-Art 2D Method

Active illumination is a prominent complement to enhance 2D face recognition and make it more robust, e.g., to spoofing attacks and low-light conditions. In the present work we show that it is possible to adopt active illumination to enhance state-of-the-art 2D face recognition approaches with 3D features, while bypassing the complicated task of 3D reconstruction. The key idea is to project over the test face a high spatial frequency pattern, which allows us to simultaneously recover real 3D information plus a standard 2D facial image. Therefore, state-of-the-art 2D face recognition solution can be transparently applied, while from the high frequency component of the input image, complementary 3D facial features are extracted. Experimental results on ND-2006 dataset show that the proposed ideas can significantly boost face recognition performance and dramatically improve the robustness to spoofing attacks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Nested Learning For Multi-Granular Tasks

Standard deep neural networks (DNNs) are commonly trained in an end-to-end fashion for specific tasks such as object recognition, face identification, or character recognition, among many examples. This specificity often leads to overconfident models that generalize poorly to samples that are not from the original training distribution. Moreover, such standard DNNs do not allow to leverage information from heterogeneously annotated training data, where for example, labels may be provided with different levels of granularity. Furthermore, DNNs do not produce results with simultaneous different levels of confidence for different levels of detail, they are most commonly an all or nothing approach. To address these challenges, we introduce the concept of nested learning: how to obtain a hierarchical representation of the input such that a coarse label can be extracted first, and sequentially refine this representation, if the sample permits, to obtain successively refined predictions, all of them with the corresponding confidence. We explicitly enforce this behavior by creating a sequence of nested information bottlenecks. Looking at the problem of nested learning from an information theory perspective, we design a network topology with two important properties. First, a sequence of low dimensional (nested) feature embeddings are enforced. Then we show how the explicit combination of nested outputs can improve both the robustness and the accuracy of finer predictions. Experimental results on Cifar-10, Cifar-100, MNIST, Fashion-MNIST, Dbpedia, and Plantvillage demonstrate that nested learning outperforms the same network trained in the standard end-to-end fashion.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Stochastic Conditional Generative Networks with Basis Decomposition

While generative adversarial networks (GANs) have revolutionized machine learning, a number of open questions remain to fully understand them and exploit their power. One of these questions is how to efficiently achieve proper diversity and sampling of the multi-mode data space. To address this, we introduce BasisGAN, a stochastic conditional multi-mode image generator. By exploiting the observation that a convolutional filter can be well approximated as a linear combination of a small set of basis elements, we learn a plug-and-played basis generator to stochastically generate basis elements, with just a few hundred of parameters, to fully embed stochasticity into convolutional filters. By sampling basis elements instead of filters, we dramatically reduce the cost of modeling the parameter space with no sacrifice on either image diversity or fidelity. To illustrate this proposed plug-and-play framework, we construct variants of BasisGAN based on state-of-the-art conditional image generation networks, and train the networks by simply plugging in a basis generator, without additional auxiliary components, hyperparameters, or training objectives. The experimental success is complemented with theoretical results indicating how the perturbations introduced by the proposed sampling of basis elements can propagate to the appearance of generated images.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Quantum Computing at the Frontiers of Biological Sciences

The search for meaningful structure in biological data has relied on cutting-edge advances in computational technology and data science methods. However, challenges arise as we push the limits of scale and complexity in biological problems. Innovation in massively parallel, classical computing hardware and algorithms continues to address many of these challenges, but there is a need to simultaneously consider new paradigms to circumvent current barriers to processing speed. Accordingly, we articulate a view towards quantum computation and quantum information science, where algorithms have demonstrated potential polynomial and exponential computational speedups in certain applications, such as machine learning. The maturation of the field of quantum computing, in hardware and algorithm development, also coincides with the growth of several collaborative efforts to address questions across length and time scales, and scientific disciplines. We use this coincidence to explore the potential for quantum computing to aid in one such endeavor: the merging of insights from genetics, genomics, neuroimaging and behavioral phenotyping. By examining joint opportunities for computational innovation across fields, we highlight the need for a common language between biological data analysis and quantum computing. Ultimately, we consider current and future prospects for the employment of quantum computing algorithms in the biological sciences.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A short-graph Fourier transform via personalized PageRank vectors

The short-time Fourier transform (STFT) is widely used to analyze the spectra of temporal signals that vary through time. Signals defined over graphs, due to their intrinsic complexity, exhibit large variations in their patterns. In this work we propose a new formulation for an STFT for signals defined over graphs. This formulation draws on recent ideas from spectral graph theory, using personalized PageRank vectors as its fundamental building block. Furthermore, this work establishes and explores the connection between local spectral graph theory and localized spectral analysis of graph signals. We accompany the presentation with synthetic and real-world examples, showing the suitability of the proposed approach.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?

Three important properties of a classification machinery are: (i) the system preserves the core information of the input data; (ii) the training examples convey information about unseen data; and (iii) the system is able to treat differently points from different classes. In this work we show that these fundamental properties are satisfied by the architecture of deep neural networks. We formally prove that these networks with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data. Similar points at the input of the network are likely to have a similar output. The theoretical analysis of deep networks here presented exploits tools used in the compressed sensing and dictionary learning literature, thereby making a formal connection between these important topics. The derived results allow drawing conclusions on the metric learning properties of the network and their relation to its structure, as well as providing bounds on the required size of the training set such that the training examples would represent faithfully the unseen data. The results are validated with state-of-the-art trained networks.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Deep Video Deblurring

Motion blur from camera shake is a major problem in videos captured by hand-held devices. Unlike single-image deblurring, video-based approaches can take advantage of the abundant information that exists across neighboring frames. As a result the best performing methods rely on aligning nearby frames. However, aligning images is a computationally expensive and fragile procedure, and methods that aggregate information must therefore be able to identify which regions have been accurately aligned and which have not, a task which requires high level scene understanding. In this work, we introduce a deep learning solution to video deblurring, where a CNN is trained end-to-end to learn how to accumulate information across frames. To train this network, we collected a dataset of real videos recorded with a high framerate camera, which we use to generate synthetic motion blur for supervision. We show that the features learned from this dataset extend to deblurring motion blur that arises due to camera shake in a wide range of videos, and compare the quality of results to a number of other baselines.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Dissimilarity-based Sparse Subset Selection

Finding an informative subset of a large collection of data points or models is at the center of many problems in computer vision, recommender systems, bio/health informatics as well as image and natural language processing. Given pairwise dissimilarities between the elements of a `source set' and a `target set,' we consider the problem of finding a subset of the source set, called representatives or exemplars, that can efficiently describe the target set. We formulate the problem as a row-sparsity regularized trace minimization problem. Since the proposed formulation is, in general, NP-hard, we consider a convex relaxation. The solution of our optimization finds representatives and the assignment of each element of the target set to each representative, hence, obtaining a clustering. We analyze the solution of our proposed optimization as a function of the regularization parameter. We show that when the two sets jointly partition into multiple groups, our algorithm finds representatives from all groups and reveals clustering of the sets. In addition, we show that the proposed framework can effectively deal with outliers. Our algorithm works with arbitrary dissimilarities, which can be asymmetric or violate the triangle inequality. To efficiently implement our algorithm, we consider an Alternating Direction Method of Multipliers (ADMM) framework, which results in quadratic complexity in the problem size. We show that the ADMM implementation allows to parallelize the algorithm, hence further reducing the computational time. Finally, by experiments on real-world datasets, we show that our proposed algorithm improves the state of the art on the two problems of scene categorization using representative images and time-series modeling and segmentation using representative~models.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Fast L1-NMF for Multiple Parametric Model Estimation

In this work we introduce a comprehensive algorithmic pipeline for multiple parametric model estimation. The proposed approach analyzes the information produced by a random sampling algorithm (e.g., RANSAC) from a machine learning/optimization perspective, using a \textit{parameterless} biclustering algorithm based on L1 nonnegative matrix factorization (L1-NMF). The proposed framework exploits consistent patterns that naturally arise during the RANSAC execution, while explicitly avoiding spurious inconsistencies. Contrarily to the main trends in the literature, the proposed technique does not impose non-intersecting parametric models. A new accelerated algorithm to compute L1-NMFs allows to handle medium-sized problems faster while also extending the usability of the algorithm to much larger datasets. This accelerated algorithm has applications in any other context where an L1-NMF is needed, beyond the biclustering approach to parameter estimation here addressed. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Fundamental Limits in Multi-image Alignment

The performance of multi-image alignment, bringing different images into one coordinate system, is critical in many applications with varied signal-to-noise ratio (SNR) conditions. A great amount of effort is being invested into developing methods to solve this problem. Several important questions thus arise, including: Which are the fundamental limits in multi-image alignment performance? Does having access to more images improve the alignment? Theoretical bounds provide a fundamental benchmark to compare methods and can help establish whether improvements can be made. In this work, we tackle the problem of finding the performance limits in image registration when multiple shifted and noisy observations are available. We derive and analyze the Cramér-Rao and Ziv-Zakai lower bounds under different statistical models for the underlying image. The accuracy of the derived bounds is experimentally assessed through a comparison to the maximum likelihood estimator. We show the existence of different behavior zones depending on the difficulty level of the problem, given by the SNR conditions of the input images. We find that increasing the number of images is only useful below a certain SNR threshold, above which the pairwise MLE estimation proves to be optimal. The analysis we present here brings further insight into the fundamental limitations of the multi-image alignment problem.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

GraphConnect: A Regularization Framework for Neural Networks

Deep neural networks have proved very successful in domains where large training sets are available, but when the number of training samples is small, their performance suffers from overfitting. Prior methods of reducing overfitting such as weight decay, Dropout and DropConnect are data-independent. This paper proposes a new method, GraphConnect, that is data-dependent, and is motivated by the observation that data of interest lie close to a manifold. The new method encourages the relationships between the learned decisions to resemble a graph representing the manifold structure. Essentially GraphConnect is designed to learn attributes that are present in data samples in contrast to weight decay, Dropout and DropConnect which are simply designed to make it more difficult to fit to random error or noise. Empirical Rademacher complexity is used to connect the generalization error of the neural network to spectral properties of the graph learned from the input data. This framework is used to show that GraphConnect is superior to weight decay. Experimental results on several benchmark datasets validate the theoretical analysis, and show that when the number of training samples is small, GraphConnect is able to significantly improve performance over weight decay.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Not Afraid of the Dark: NIR-VIS Face Recognition via Cross-spectral Hallucination and Low-rank Embedding

Surveillance cameras today often capture NIR (near infrared) images in low-light environments. However, most face datasets accessible for training and verification are only collected in the VIS (visible light) spectrum. It remains a challenging problem to match NIR to VIS face images due to the different light spectrum. Recently, breakthroughs have been made for VIS face recognition by applying deep learning on a huge amount of labeled VIS face samples. The same deep learning approach cannot be simply applied to NIR face recognition for two main reasons: First, much limited NIR face images are available for training compared to the VIS spectrum. Second, face galleries to be matched are mostly available only in the VIS spectrum. In this paper, we propose an approach to extend the deep learning breakthrough for VIS face recognition to the NIR spectrum, without retraining the underlying deep models that see only VIS faces. Our approach consists of two core components, cross-spectral hallucination and low-rank embedding, to optimize respectively input and output of a VIS deep model for cross-spectral face recognition. Cross-spectral hallucination produces VIS faces from NIR images through a deep learning approach. Low-rank embedding restores a low-rank structure for faces deep features across both NIR and VIS spectrum. We observe that it is often equally effective to perform hallucination to input NIR images or low-rank embedding to output deep features for a VIS deep model for cross-spectral recognition. When hallucination and low-rank embedding are deployed together, we observe significant further improvement; we obtain state-of-the-art accuracy on the CASIA NIR-VIS v2.0 benchmark, without the need at all to re-train the recognition system.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Self-learning Scene-specific Pedestrian Detectors using a Progressive Latent Model

In this paper, a self-learning approach is proposed towards solving scene-specific pedestrian detection problem without any human' annotation involved. The self-learning approach is deployed as progressive steps of object discovery, object enforcement, and label propagation. In the learning procedure, object locations in each frame are treated as latent variables that are solved with a progressive latent model (PLM). Compared with conventional latent models, the proposed PLM incorporates a spatial regularization term to reduce ambiguities in object proposals and to enforce object localization, and also a graph-based label propagation to discover harder instances in adjacent frames. With the difference of convex (DC) objective functions, PLM can be efficiently optimized with a concave-convex programming and thus guaranteeing the stability of self-learning. Extensive experiments demonstrate that even without annotation the proposed self-learning approach outperforms weakly supervised learning approaches, while achieving comparable performance with transfer learning and fully supervised approaches.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Compressed Nonnegative Matrix Factorization is Fast and Accurate

Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences. To address this, in this work we propose to use structured random compression, that is, random projections that exploit the data structure, for two NMF variants: classical and separable. In separable NMF (SNMF) the left factors are a subset of the columns of the input matrix. We present suitable formulations for each problem, dealing with different representative algorithms within each one. We show that the resulting compressed techniques are faster than their uncompressed variants, vastly reduce memory demands, and do not encompass any significant deterioration in performance. The proposed structured random projections for SNMF allow to deal with arbitrarily shaped large matrices, beyond the standard limit of tall-and-skinny matrices, granting access to very efficient computations in this general setting. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples, showing the suitability of the proposed approaches.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Data Representation using the Weyl Transform

The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform has a high degree of symmetry with respect to a large group of multiscale transformations, which allows compact yet discriminative representations to be obtained by pooling coefficients. The effectiveness of the Weyl transform is demonstrated through the example of textured image classification.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Geometry-aware Deep Transform

Many recent efforts have been devoted to designing sophisticated deep learning structures, obtaining revolutionary results on benchmark datasets. The success of these deep learning methods mostly relies on an enormous volume of labeled training samples to learn a huge number of parameters in a network; therefore, understanding the generalization ability of a learned deep network cannot be overlooked, especially when restricted to a small training set, which is the case for many applications. In this paper, we propose a novel deep learning objective formulation that unifies both the classification and metric learning criteria. We then introduce a geometry-aware deep transform to enable a non-linear discriminative and robust feature transform, which shows competitive performance on small training sets for both synthetic and real-world data. We further support the proposed framework with a formal $(K,ε)$-robustness analysis.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Graph Matching: Relax at Your Own Risk

Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Hand-held Video Deblurring via Efficient Fourier Aggregation

Videos captured with hand-held cameras often suffer from a significant amount of blur, mainly caused by the inevitable natural tremor of the photographer's hand. In this work, we present an algorithm that removes blur due to camera shake by combining information in the Fourier domain from nearby frames in a video. The dynamic nature of typical videos with the presence of multiple moving objects and occlusions makes this problem of camera shake removal extremely challenging, in particular when low complexity is needed. Given an input video frame, we first create a consistent registered version of temporally adjacent frames. Then, the set of consistently registered frames is block-wise fused in the Fourier domain with weights depending on the Fourier spectrum magnitude. The method is motivated from the physiological fact that camera shake blur has a random nature and therefore, nearby video frames are generally blurred differently. Experiments with numerous videos recorded in the wild, along with extensive comparisons, show that the proposed algorithm achieves state-of-the-art results while at the same time being much faster than its competitors.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

On the Stability of Deep Networks

In this work we study the properties of deep neural networks (DNN) with random weights. We formally prove that these networks perform a distance-preserving embedding of the data. Based on this we then draw conclusions on the size of the training data and the networks' structure. A longer version of this paper with more results and details can be found in (Giryes et al., 2015). In particular, we formally prove in the longer version that DNN with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Random Forests Can Hash

Hash codes are a very efficient data representation needed to be able to cope with the ever growing amounts of data. We introduce a random forest semantic hashing scheme with information-theoretic code aggregation, showing for the first time how random forest, a technique that together with deep learning have shown spectacular results in classification, can also be extended to large-scale retrieval. Traditional random forest fails to enforce the consistency of hashes generated from each tree for the same class data, i.e., to preserve the underlying similarity, and it also lacks a principled way for code aggregation across trees. We start with a simple hashing scheme, where independently trained random trees in a forest are acting as hashing functions. We the propose a subspace model as the splitting function, and show that it enforces the hash consistency in a tree for data from the same class. We also introduce an information-theoretic approach for aggregating codes of individual trees into a single hash code, producing a near-optimal unique hash for each class. Experiments on large-scale public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art hashing methods for retrieval tasks.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Removing Camera Shake via Weighted Fourier Burst Accumulation

Numerous recent approaches attempt to remove image blur due to camera shake, either with one or multiple input images, by explicitly solving an inverse and inherently ill-posed deconvolution problem. If the photographer takes a burst of images, a modality available in virtually all modern digital cameras, we show that it is possible to combine them to get a clean sharp version. This is done without explicitly solving any blur estimation and subsequent inverse problem. The proposed algorithm is strikingly simple: it performs a weighted average in the Fourier domain, with weights depending on the Fourier spectrum magnitude. The method can be seen as a generalization of the align and average procedure, with a weighted average, motivated by hand-shake physiology and theoretically supported, taking place in the Fourier domain. The method's rationale is that camera shake has a random nature and therefore each image in the burst is generally blurred differently. Experiments with real camera data, and extensive comparisons, show that the proposed Fourier Burst Accumulation (FBA) algorithm achieves state-of-the-art results an order of magnitude faster, with simplicity for on-board implementation on camera phones. Finally, we also present experiments in real high dynamic range (HDR) scenes, showing how the method can be straightforwardly extended to HDR photography.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

A Bi-clustering Framework for Consensus Problems

We consider grouping as a general characterization for problems such as clustering, community detection in networks, and multiple parametric model estimation. We are interested in merging solutions from different grouping algorithms, distilling all their good qualities into a consensus solution. In this paper, we propose a bi-clustering framework and perspective for reaching consensus in such grouping problems. In particular, this is the first time that the task of finding/fitting multiple parametric models to a dataset is formally posed as a consensus problem. We highlight the equivalence of these tasks and establish the connection with the computational Gestalt program, that seeks to provide a psychologically-inspired detection theory for visual events. We also present a simple but powerful bi-clustering algorithm, specially tuned to the nature of the problem we address, though general enough to handle many different instances inscribed within our characterization. The presentation is accompanied with diverse and extensive experimental results in clustering, community detection, and multiple parametric model estimation in image processing applications.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Learning Transformations for Classification Forests

This work introduces a transformation-based learner model for classification forests. The weak learner at each split node plays a crucial role in a classification tree. We propose to optimize the splitting objective by learning a linear transformation on subspaces using nuclear norm as the optimization criteria. The learned linear transformation restores a low-rank structure for data from the same class, and, at the same time, maximizes the separation between different classes, thereby improving the performance of the split function. Theoretical and experimental results support the proposed framework.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Learning Transformations for Clustering and Classification

A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the literature to partition such high-dimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models. We propose to address this by learning a linear transformation on subspaces using matrix rank, via its convex surrogate nuclear norm, as the optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a a maximally separated structure for data from different subspaces. In this way, we reduce variations within subspaces, and increase separation between subspaces for a more robust subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. Basic theoretical results here presented help to further support the underlying framework. To exploit the low-rank structures of the transformed subspaces, we further introduce a fast subspace clustering technique, which efficiently combines robust PCA with sparse modeling. When class labels are present at the training stage, we show this low-rank transformation framework also significantly enhances classification performance. Extensive experiments using public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Low-Cost Compressive Sensing for Color Video and Depth

A simple and inexpensive (low-power and low-bandwidth) modification is made to a conventional off-the-shelf color video camera, from which we recover {multiple} color frames for each of the original measured frames, and each of the recovered frames can be focused at a different depth. The recovery of multiple frames for each measured frame is made possible via high-speed coding, manifested via translation of a single coded aperture; the inexpensive translation is constituted by mounting the binary code on a piezoelectric device. To simultaneously recover depth information, a {liquid} lens is modulated at high speed, via a variable voltage. Consequently, during the aforementioned coding process, the liquid lens allows the camera to sweep the focus through multiple depths. In addition to designing and implementing the camera, fast recovery is achieved by an anytime algorithm exploiting the group-sparsity of wavelet/DCT coefficients.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

On spectral properties for graph matching and graph isomorphism problems

Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer. In this paper we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Sparse similarity-preserving hashing

In recent years, a lot of attention has been devoted to efficient nearest neighbor search by means of similarity-preserving hashing. One of the plights of existing hashing techniques is the intrinsic trade-off between performance and computational complexity: while longer hash codes allow for lower false positive rates, it is very difficult to increase the embedding dimensionality without incurring in very high false negatives rates or prohibiting computational costs. In this paper, we propose a way to overcome this limitation by enforcing the hash codes to be sparse. Sparse high-dimensional codes enjoy from the low false positive rates typical of long hashes, while keeping the false negative rates similar to those of a shorter dense hashing scheme with equal number of degrees of freedom. We use a tailored feed-forward neural network for the hashing function. Extensive experimental evaluation involving visual and multi-modal data shows the benefits of the proposed method.

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preprint2013arXiv

Adaptive Temporal Compressive Sensing for Video

This paper introduces the concept of adaptive temporal compressive sensing (CS) for video. We propose a CS algorithm to adapt the compression ratio based on the scene's temporal complexity, computed from the compressed data, without compromising the quality of the reconstructed video. The temporal adaptivity is manifested by manipulating the integration time of the camera, opening the possibility to real-time implementation. The proposed algorithm is a generalized temporal CS approach that can be incorporated with a diverse set of existing hardware systems.

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preprint2013arXiv

Domain-invariant Face Recognition using Learned Low-rank Transformation

We present a low-rank transformation approach to compensate for face variations due to changes in visual domains, such as pose and illumination. The key idea is to learn discriminative linear transformations for face images using matrix rank as the optimization criteria. The learned linear transformations restore a shared low-rank structure for faces from the same subject, and, at the same time, force a high-rank structure for faces from different subjects. In this way, among the transformed faces, we reduce variations caused by domain changes within the classes, and increase separations between the classes for better face recognition across domains. Extensive experiments using public datasets are presented to demonstrate the effectiveness of our approach for face recognition across domains. The potential of the approach for feature extraction in generic object recognition and coded aperture design are discussed as well.

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preprint2013arXiv

Learning Robust Subspace Clustering

We propose a low-rank transformation-learning framework to robustify subspace clustering. Many high-dimensional data, such as face images and motion sequences, lie in a union of low-dimensional subspaces. The subspace clustering problem has been extensively studied in the literature to partition such high-dimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models. We propose to address this by learning a linear transformation on subspaces using matrix rank, via its convex surrogate nuclear norm, as the optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a high-rank structure for data from different subspaces. In this way, we reduce variations within the subspaces, and increase separations between the subspaces for more accurate subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. To exploit the low-rank structures of the transformed subspaces, we further introduce a subspace clustering technique, called Robust Sparse Subspace Clustering, which efficiently combines robust PCA with sparse modeling. We also discuss the online learning of the transformation, and learning of the transformation while simultaneously reducing the data dimensionality. Extensive experiments using public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art subspace clustering methods.

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preprint2013arXiv

Robust Multimodal Graph Matching: Sparse Coding Meets Graph Matching

Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsity-related techniques. We cast the problem, resembling group or collaborative sparsity formulations, as a non-smooth convex optimization problem that can be efficiently solved using augmented Lagrangian techniques. The method can deal with weighted or unweighted graphs, as well as multimodal data, where different graphs represent different types of data. The proposed approach is also naturally integrated with collaborative graph inference techniques, solving general network inference problems where the observed variables, possibly coming from different modalities, are not in correspondence. The algorithm is tested and compared with state-of-the-art graph matching techniques in both synthetic and real graphs. We also present results on multimodal graphs and applications to collaborative inference of brain connectivity from alignment-free functional magnetic resonance imaging (fMRI) data. The code is publicly available.

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preprint2013arXiv

Video Human Segmentation using Fuzzy Object Models and its Application to Body Pose Estimation of Toddlers for Behavior Studies

Video object segmentation is a challenging problem due to the presence of deformable, connected, and articulated objects, intra- and inter-object occlusions, object motion, and poor lighting. Some of these challenges call for object models that can locate a desired object and separate it from its surrounding background, even when both share similar colors and textures. In this work, we extend a fuzzy object model, named cloud system model (CSM), to handle video segmentation, and evaluate it for body pose estimation of toddlers at risk of autism. CSM has been successfully used to model the parts of the brain (cerebrum, left and right brain hemispheres, and cerebellum) in order to automatically locate and separate them from each other, the connected brain stem, and the background in 3D MR-images. In our case, the objects are articulated parts (2D projections) of the human body, which can deform, cause self-occlusions, and move along the video. The proposed CSM extension handles articulation by connecting the individual clouds, body parts, of the system using a 2D stickman model. The stickman representation naturally allows us to extract 2D body pose measures of arm asymmetry patterns during unsupported gait of toddlers, a possible behavioral marker of autism. The results show that our method can provide insightful knowledge to assist the specialist's observations during real in-clinic assessments.

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preprint2012arXiv

A Complete System for Candidate Polyps Detection in Virtual Colonoscopy

Computer tomographic colonography, combined with computer-aided detection, is a promising emerging technique for colonic polyp analysis. We present a complete pipeline for polyp detection, starting with a simple colon segmentation technique that enhances polyps, followed by an adaptive-scale candidate polyp delineation and classification based on new texture and geometric features that consider both the information in the candidate polyp location and its immediate surrounding area. The proposed system is tested with ground truth data, including flat and small polyps which are hard to detect even with optical colonoscopy. For polyps larger than 6mm in size we achieve 100% sensitivity with just 0.9 false positives per case, and for polyps larger than 3mm in size we achieve 93% sensitivity with 2.8 false positives per case.

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preprint2012arXiv

Are You Imitating Me? Unsupervised Sparse Modeling for Group Activity Analysis from a Single Video

A framework for unsupervised group activity analysis from a single video is here presented. Our working hypothesis is that human actions lie on a union of low-dimensional subspaces, and thus can be efficiently modeled as sparse linear combinations of atoms from a learned dictionary representing the action's primitives. Contrary to prior art, and with the primary goal of spatio-temporal action grouping, in this work only one single video segment is available for both unsupervised learning and analysis without any prior training information. After extracting simple features at a single spatio-temporal scale, we learn a dictionary for each individual in the video during each short time lapse. These dictionaries allow us to compare the individuals' actions by producing an affinity matrix which contains sufficient discriminative information about the actions in the scene leading to grouping with simple and efficient tools. With diverse publicly available real videos, we demonstrate the effectiveness of the proposed framework and its robustness to cluttered backgrounds, changes of human appearance, and action variability.

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preprint2012arXiv

Computer vision tools for the non-invasive assessment of autism-related behavioral markers

The early detection of developmental disorders is key to child outcome, allowing interventions to be initiated that promote development and improve prognosis. Research on autism spectrum disorder (ASD) suggests behavioral markers can be observed late in the first year of life. Many of these studies involved extensive frame-by-frame video observation and analysis of a child's natural behavior. Although non-intrusive, these methods are extremely time-intensive and require a high level of observer training; thus, they are impractical for clinical and large population research purposes. Diagnostic measures for ASD are available for infants but are only accurate when used by specialists experienced in early diagnosis. This work is a first milestone in a long-term multidisciplinary project that aims at helping clinicians and general practitioners accomplish this early detection/measurement task automatically. We focus on providing computer vision tools to measure and identify ASD behavioral markers based on components of the Autism Observation Scale for Infants (AOSI). In particular, we develop algorithms to measure three critical AOSI activities that assess visual attention. We augment these AOSI activities with an additional test that analyzes asymmetrical patterns in unsupported gait. The first set of algorithms involves assessing head motion by tracking facial features, while the gait analysis relies on joint foreground segmentation and 2D body pose estimation in video. We show results that provide insightful knowledge to augment the clinician's behavioral observations obtained from real in-clinic assessments.

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preprint2012arXiv

L1 Splines for Robust, Simple, and Fast Smoothing of Grid Data

Splines are a popular and attractive way of smoothing noisy data. Computing splines involves minimizing a functional which is a linear combination of a fitting term and a regularization term. The former is classically computed using a (weighted) L2 norm while the latter ensures smoothness. Thus, when dealing with grid data, the optimization can be solved very efficiently using the DCT. In this work we propose to replace the L2 norm in the fitting term with an L1 norm, leading to automatic robustness to outliers. To solve the resulting minimization problem we propose an extremely simple and efficient numerical scheme based on split-Bregman iteration combined with DCT. Experimental validation shows the high-quality results obtained in short processing times.

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preprint2012arXiv

Learning efficient sparse and low rank models

Parsimony, including sparsity and low rank, has been shown to successfully model data in numerous machine learning and signal processing tasks. Traditionally, such modeling approaches rely on an iterative algorithm that minimizes an objective function with parsimony-promoting terms. The inherently sequential structure and data-dependent complexity and latency of iterative optimization constitute a major limitation in many applications requiring real-time performance or involving large-scale data. Another limitation encountered by these modeling techniques is the difficulty of their inclusion in discriminative learning scenarios. In this work, we propose to move the emphasis from the model to the pursuit algorithm, and develop a process-centric view of parsimonious modeling, in which a learned deterministic fixed-complexity pursuit process is used in lieu of iterative optimization. We show a principled way to construct learnable pursuit process architectures for structured sparse and robust low rank models, derived from the iteration of proximal descent algorithms. These architectures learn to approximate the exact parsimonious representation at a fraction of the complexity of the standard optimization methods. We also show that appropriate training regimes allow to naturally extend parsimonious models to discriminative settings. State-of-the-art results are demonstrated on several challenging problems in image and audio processing with several orders of magnitude speedup compared to the exact optimization algorithms.

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preprint2012arXiv

Learning Efficient Structured Sparse Models

We present a comprehensive framework for structured sparse coding and modeling extending the recent ideas of using learnable fast regressors to approximate exact sparse codes. For this purpose, we develop a novel block-coordinate proximal splitting method for the iterative solution of hierarchical sparse coding problems, and show an efficient feed forward architecture derived from its iteration. This architecture faithfully approximates the exact structured sparse codes with a fraction of the complexity of the standard optimization methods. We also show that by using different training objective functions, learnable sparse encoders are no longer restricted to be mere approximants of the exact sparse code for a pre-given dictionary, as in earlier formulations, but can be rather used as full-featured sparse encoders or even modelers. A simple implementation shows several orders of magnitude speedup compared to the state-of-the-art at minimal performance degradation, making the proposed framework suitable for real time and large-scale applications.

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preprint2012arXiv

Learning Robust Low-Rank Representations

In this paper we present a comprehensive framework for learning robust low-rank representations by combining and extending recent ideas for learning fast sparse coding regressors with structured non-convex optimization techniques. This approach connects robust principal component analysis (RPCA) with dictionary learning techniques and allows its approximation via trainable encoders. We propose an efficient feed-forward architecture derived from an optimization algorithm designed to exactly solve robust low dimensional projections. This architecture, in combination with different training objective functions, allows the regressors to be used as online approximants of the exact offline RPCA problem or as RPCA-based neural networks. Simple modifications of these encoders can handle challenging extensions, such as the inclusion of geometric data transformations. We present several examples with real data from image, audio, and video processing. When used to approximate RPCA, our basic implementation shows several orders of magnitude speedup compared to the exact solvers with almost no performance degradation. We show the strength of the inclusion of learning to the RPCA approach on a music source separation application, where the encoders outperform the exact RPCA algorithms, which are already reported to produce state-of-the-art results on a benchmark database. Our preliminary implementation on an iPad shows faster-than-real-time performance with minimal latency.

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preprint2012arXiv

Semi-Supervised Single- and Multi-Domain Regression with Multi-Domain Training

We address the problems of multi-domain and single-domain regression based on distinct and unpaired labeled training sets for each of the domains and a large unlabeled training set from all domains. We formulate these problems as a Bayesian estimation with partial knowledge of statistical relations. We propose a worst-case design strategy and study the resulting estimators. Our analysis explicitly accounts for the cardinality of the labeled sets and includes the special cases in which one of the labeled sets is very large or, in the other extreme, completely missing. We demonstrate our estimators in the context of removing expressions from facial images and in the context of audio-visual word recognition, and provide comparisons to several recently proposed multi-modal learning algorithms.

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preprint2012arXiv

Task-Driven Adaptive Statistical Compressive Sensing of Gaussian Mixture Models

A framework for adaptive and non-adaptive statistical compressive sensing is developed, where a statistical model replaces the standard sparsity model of classical compressive sensing. We propose within this framework optimal task-specific sensing protocols specifically and jointly designed for classification and reconstruction. A two-step adaptive sensing paradigm is developed, where online sensing is applied to detect the signal class in the first step, followed by a reconstruction step adapted to the detected class and the observed samples. The approach is based on information theory, here tailored for Gaussian mixture models (GMMs), where an information-theoretic objective relationship between the sensed signals and a representation of the specific task of interest is maximized. Experimental results using synthetic signals, Landsat satellite attributes, and natural images of different sizes and with different noise levels show the improvements achieved using the proposed framework when compared to more standard sensing protocols. The underlying formulation can be applied beyond GMMs, at the price of higher mathematical and computational complexity.

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preprint2011arXiv

A convex model for non-negative matrix factorization and dimensionality reduction on physical space

A collaborative convex framework for factoring a data matrix $X$ into a non-negative product $AS$, with a sparse coefficient matrix $S$, is proposed. We restrict the columns of the dictionary matrix $A$ to coincide with certain columns of the data matrix $X$, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use $l_{1,\infty}$ regularization to select the dictionary from the data and show this leads to an exact convex relaxation of $l_0$ in the case of distinct noise free data. We also show how to relax the restriction-to-$X$ constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in $X$. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.

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preprint2011arXiv

Blind Compressed Sensing Over a Structured Union of Subspaces

This paper addresses the problem of simultaneous signal recovery and dictionary learning based on compressive measurements. Multiple signals are analyzed jointly, with multiple sensing matrices, under the assumption that the unknown signals come from a union of a small number of disjoint subspaces. This problem is important, for instance, in image inpainting applications, in which the multiple signals are constituted by (incomplete) image patches taken from the overall image. This work extends standard dictionary learning and block-sparse dictionary optimization, by considering compressive measurements, e.g., incomplete data). Previous work on blind compressed sensing is also generalized by using multiple sensing matrices and relaxing some of the restrictions on the learned dictionary. Drawing on results developed in the context of matrix completion, it is proven that both the dictionary and signals can be recovered with high probability from compressed measurements. The solution is unique up to block permutations and invertible linear transformations of the dictionary atoms. The recovery is contingent on the number of measurements per signal and the number of signals being sufficiently large; bounds are derived for these quantities. In addition, this paper presents a computationally practical algorithm that performs dictionary learning and signal recovery, and establishes conditions for its convergence to a local optimum. Experimental results for image inpainting demonstrate the capabilities of the method.

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preprint2011arXiv

C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework

Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an L1-regularized linear regression problem, commonly referred to as Lasso or Basis Pursuit. In this work we combine the sparsity-inducing property of the Lasso model at the individual feature level, with the block-sparsity property of the Group Lasso model, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the Hierarchical Lasso (HiLasso), which shows important practical modeling advantages. We then extend this approach to the collaborative case, where a set of simultaneously coded signals share the same sparsity pattern at the higher (group) level, but not necessarily at the lower (inside the group) level, obtaining the collaborative HiLasso model (C-HiLasso). Such signals then share the same active groups, or classes, but not necessarily the same active set. This model is very well suited for applications such as source identification and separation. An efficient optimization procedure, which guarantees convergence to the global optimum, is developed for these new models. The underlying presentation of the new framework and optimization approach is complemented with experimental examples and theoretical results regarding recovery guarantees for the proposed models.

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preprint2011arXiv

Low-rank data modeling via the Minimum Description Length principle

Robust low-rank matrix estimation is a topic of increasing interest, with promising applications in a variety of fields, from computer vision to data mining and recommender systems. Recent theoretical results establish the ability of such data models to recover the true underlying low-rank matrix when a large portion of the measured matrix is either missing or arbitrarily corrupted. However, if low rank is not a hypothesis about the true nature of the data, but a device for extracting regularity from it, no current guidelines exist for choosing the rank of the estimated matrix. In this work we address this problem by means of the Minimum Description Length (MDL) principle -- a well established information-theoretic approach to statistical inference -- as a guideline for selecting a model for the data at hand. We demonstrate the practical usefulness of our formal approach with results for complex background extraction in video sequences.

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preprint2011arXiv

Online Adaptive Statistical Compressed Sensing of Gaussian Mixture Models

A framework of online adaptive statistical compressed sensing is introduced for signals following a mixture model. The scheme first uses non-adaptive measurements, from which an online decoding scheme estimates the model selection. As soon as a candidate model has been selected, an optimal sensing scheme for the selected model continues to apply. The final signal reconstruction is calculated from the ensemble of both the non-adaptive and the adaptive measurements. For signals generated from a Gaussian mixture model, the online adaptive sensing algorithm is given and its performance is analyzed. On both synthetic and real image data, the proposed adaptive scheme considerably reduces the average reconstruction error with respect to standard statistical compressed sensing that uses fully random measurements, at a marginally increased computational complexity.

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preprint2011arXiv

Statistical Compressed Sensing of Gaussian Mixture Models

A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximum a posteriori expectation-maximization algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.

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preprint2010arXiv

Collaborative Hierarchical Sparse Modeling

Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is done by solving an l_1-regularized linear regression problem, usually called Lasso. In this work we first combine the sparsity-inducing property of the Lasso model, at the individual feature level, with the block-sparsity property of the group Lasso model, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the hierarchical Lasso, which shows important practical modeling advantages. We then extend this approach to the collaborative case, where a set of simultaneously coded signals share the same sparsity pattern at the higher (group) level but not necessarily at the lower one. Signals then share the same active groups, or classes, but not necessarily the same active set. This is very well suited for applications such as source separation. An efficient optimization procedure, which guarantees convergence to the global optimum, is developed for these new models. The underlying presentation of the new framework and optimization approach is complemented with experimental examples and preliminary theoretical results.

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preprint2010arXiv

Collaborative Sources Identification in Mixed Signals via Hierarchical Sparse Modeling

A collaborative framework for detecting the different sources in mixed signals is presented in this paper. The approach is based on C-HiLasso, a convex collaborative hierarchical sparse model, and proceeds as follows. First, we build a structured dictionary for mixed signals by concatenating a set of sub-dictionaries, each one of them learned to sparsely model one of a set of possible classes. Then, the coding of the mixed signal is performed by efficiently solving a convex optimization problem that combines standard sparsity with group and collaborative sparsity. The present sources are identified by looking at the sub-dictionaries automatically selected in the coding. The collaborative filtering in C-HiLasso takes advantage of the temporal/spatial redundancy in the mixed signals, letting collections of samples collaborate in identifying the classes, while allowing individual samples to have different internal sparse representations. This collaboration is critical to further stabilize the sparse representation of signals, in particular the class/sub-dictionary selection. The internal sparsity inside the sub-dictionaries, as naturally incorporated by the hierarchical aspects of C-HiLasso, is critical to make the model consistent with the essence of the sub-dictionaries that have been trained for sparse representation of each individual class. We present applications from speaker and instrument identification and texture separation. In the case of audio signals, we use sparse modeling to describe the short-term power spectrum envelopes of harmonic sounds. The proposed pitch independent method automatically detects the number of sources on a recording.

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preprint2010arXiv

Efficient Matrix Completion with Gaussian Models

A general framework based on Gaussian models and a MAP-EM algorithm is introduced in this paper for solving matrix/table completion problems. The numerical experiments with the standard and challenging movie ratings data show that the proposed approach, based on probably one of the simplest probabilistic models, leads to the results in the same ballpark as the state-of-the-art, at a lower computational cost.

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preprint2010arXiv

Online Learning for Matrix Factorization and Sparse Coding

Sparse coding--that is, modelling data vectors as sparse linear combinations of basis elements--is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing, non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed and optimization for both small and large datasets.

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preprint2010arXiv

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.

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preprint2010arXiv

Sparse coding and dictionary learning based on the MDL principle

The power of sparse signal coding with learned dictionaries has been demonstrated in a variety of applications and fields, from signal processing to statistical inference and machine learning. However, the statistical properties of these models, such as underfitting or overfitting given sets of data, are still not well characterized in the literature. This work aims at filling this gap by means of the Minimum Description Length (MDL) principle -- a well established information-theoretic approach to statistical inference. The resulting framework derives a family of efficient sparse coding and modeling (dictionary learning) algorithms, which by virtue of the MDL principle, are completely parameter free. Furthermore, such framework allows to incorporate additional prior information in the model, such as Markovian dependencies, in a natural way. We demonstrate the performance of the proposed framework with results for image denoising and classification tasks.

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preprint2010arXiv

Statistical Compressive Sensing of Gaussian Mixture Models

A new framework of compressive sensing (CS), namely statistical compressive sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution and achieving accurate reconstruction on average, is introduced. For signals following a Gaussian distribution, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS, where N is the signal dimension, and with an optimal decoder implemented with linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the k-best term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the k-best term approximation with probability one, and the bound constant can be efficiently calculated. For signals following Gaussian mixture models, SCS with a piecewise linear decoder is introduced and shown to produce for real images better results than conventional CS based on sparse models.

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preprint2010arXiv

Universal Regularizers For Robust Sparse Coding and Modeling

Sparse data models, where data is assumed to be well represented as a linear combination of a few elements from a dictionary, have gained considerable attention in recent years, and their use has led to state-of-the-art results in many signal and image processing tasks. It is now well understood that the choice of the sparsity regularization term is critical in the success of such models. Based on a codelength minimization interpretation of sparse coding, and using tools from universal coding theory, we propose a framework for designing sparsity regularization terms which have theoretical and practical advantages when compared to the more standard l0 or l1 ones. The presentation of the framework and theoretical foundations is complemented with examples that show its practical advantages in image denoising, zooming and classification.

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