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Marcin Mazur

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Computer Vision Machine Learning math.NT math.RA Artificial Intelligence math-ph math.AG math.MG math.MP

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preprint2026arXiv

SoftSAE: Dynamic Top-K Selection for Adaptive Sparse Autoencoders

Sparse Autoencoders (SAEs) have become an important tool in mechanistic interpretability, helping to analyze internal representations in both Large Language Models (LLMs) and Vision Transformers (ViTs). By decomposing polysemantic activations into sparse sets of monosemantic features, SAEs aim to translate neural network computations into human-understandable concepts. However, common architectures such as TopK SAEs rely on a fixed sparsity level. They enforce the same number of active features (K) across all inputs, ignoring the varying complexity of real-world data. Natural data often lies on manifolds with varying local intrinsic dimensionality, meaning the number of relevant factors can change significantly across samples. This suggests that a fixed sparsity level is not optimal. Simple inputs may require only a few features, while more complex ones need more expressive representations. Using a constant K can therefore introduce noise in simple cases or miss important structure in more complex ones. To address this issue, we propose SoftSAE, a sparse autoencoder with a Dynamic Top-K selection mechanism. Our method uses a differentiable Soft Top-K operator to learn an input-dependent sparsity level k. This allows the model to adjust the number of active features based on the complexity of each input. As a result, the representation better matches the structure of the data, and the explanation length reflects the amount of information in the input. Experimental results confirm that SoftSAE not only finds meaningful features, but also selects the right number of features for each concept. The source code is available at: https://github.com/St0pien/SoftSAE.

preprint2021arXiv

HyperPocket: Generative Point Cloud Completion

Scanning real-life scenes with modern registration devices typically give incomplete point cloud representations, mostly due to the limitations of the scanning process and 3D occlusions. Therefore, completing such partial representations remains a fundamental challenge of many computer vision applications. Most of the existing approaches aim to solve this problem by learning to reconstruct individual 3D objects in a synthetic setup of an uncluttered environment, which is far from a real-life scenario. In this work, we reformulate the problem of point cloud completion into an object hallucination task. Thus, we introduce a novel autoencoder-based architecture called HyperPocket that disentangles latent representations and, as a result, enables the generation of multiple variants of the completed 3D point clouds. We split point cloud processing into two disjoint data streams and leverage a hypernetwork paradigm to fill the spaces, dubbed pockets, that are left by the missing object parts. As a result, the generated point clouds are not only smooth but also plausible and geometrically consistent with the scene. Our method offers competitive performances to the other state-of-the-art models, and it enables a~plethora of novel applications.

preprint2020arXiv

Generative models with kernel distance in data space

Generative models dealing with modeling a~joint data distribution are generally either autoencoder or GAN based. Both have their pros and cons, generating blurry images or being unstable in training or prone to mode collapse phenomenon, respectively. The objective of this paper is to construct a~model situated between above architectures, one that does not inherit their main weaknesses. The proposed LCW generator (Latent Cramer-Wold generator) resembles a classical GAN in transforming Gaussian noise into data space. What is of utmost importance, instead of a~discriminator, LCW generator uses kernel distance. No adversarial training is utilized, hence the name generator. It is trained in two phases. First, an autoencoder based architecture, using kernel measures, is built to model a manifold of data. We propose a Latent Trick mapping a Gaussian to latent in order to get the final model. This results in very competitive FID values.

preprint2019arXiv

Cramer-Wold AutoEncoder

We propose a new generative model, Cramer-Wold Autoencoder (CWAE). Following WAE, we directly encourage normality of the latent space. Our paper uses also the recent idea from Sliced WAE (SWAE) model, which uses one-dimensional projections as a method of verifying closeness of two distributions. The crucial new ingredient is the introduction of a new (Cramer-Wold) metric in the space of densities, which replaces the Wasserstein metric used in SWAE. We show that the Cramer-Wold metric between Gaussian mixtures is given by a simple analytic formula, which results in the removal of sampling necessary to estimate the cost function in WAE and SWAE models. As a consequence, while drastically simplifying the optimization procedure, CWAE produces samples of a matching perceptual quality to other SOTA models.

preprint2014arXiv

Generators of maximal orders

Let R be the ring of algebraic integers in a number field K and let L be a maximal order in a semisimple K-algebra B. Building on our previous work, we compute the smallest number of algebra generators of L considered as an R-algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.

preprint2012arXiv

Representations of analytic functions as infinite products and their application to numerical computations

Let $D$ be an open disk of radius $\le 1$ in $\mathbb C$, and let $(ε_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f: D \to \mathbb C$ without zeros in $D$, there exists a unique sequence $(α_n)$ of complex numbers such that $f(z) = f(0)\prod_{n=1}^{\infty} (1+ε_nz^n)^{α_n}$ for every $z \in D$. From this representation we obtain a numerical method for calculating products of the form $\prod_{p \text{prime}} f(1/p)$ provided $f(0)=1$ and $f'(0) = 0$; our method generalizes a well known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $π^{-1/2}\prod_{p \text{prime}} \sqrt{p^2-p}\ln(p/(p-1))$. From the properties of the exponents $α_n$, we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every $n \times n$ integral matrix $A$, every prime number $p$, and every positive integer $k$ we have $\text{tr} A^{p^k} \equiv \text{tr} A^{p^{k-1}} (\text{mod}\,{p^k})$.

preprint2011arXiv

On the conjectures of Atiyah and Sutcliffe

Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and Sutcliffe stated three intriguing conjectures about this determinant. They provided compelling numerical evidence for the conjectures and an interesting physical interpretation of the determinant. The first conjecture asserts that the determinant never vanishes, the second states that its absolute value is at least one, and the third says that $|D(x_1,..., x_n)|^{n-2}\geq \prod_{i=1}^n |D(x_1,..., x_{i-1},x_{i+1},..., x_n)|$. Despite their simple formulation, these conjectures appear to be notoriously difficult. Let $D_n$ denote the Atiyah determinant evaluated at the vertices of a regular $n-$gon. We prove that $\lim_{n\to \infty} \frac{\ln D_n}{n^2}= \frac{7ζ(3)}{2π^2}-\frac{\ln 2}{2}=0.07970479...$ and establish the second conjecture in this case. Furthermore, we prove the second conjecture for vertices of a convex quadrilateral and the third conjecture for vertices of an inscribed quadrilateral.

preprint2010arXiv

On the smallest number of generators and the probability of generating an algebra

In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free $R$-module of finite rank. We develop a technique to compute the smallest number of generators of $A$. For example, we prove that the ring $M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. For a given positive integer $m$, we define the density of the set of all ordered $m$-tuples of elements of $A$ which generate it as an $R$-algebra. We express this density as a certain infinite product over the maximal ideals of $R$, and we interpret the resulting formula probabilistically. For example, we show that the probability that 2 random $3\times 3$ matrices generate the ring $M_3(\mathbb{Z})$ is equal to $(ζ(2)^2 ζ(3))^{-1}$, where $ζ$ is the Riemann zeta-function.