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Published work

26 published item(s)

preprint2026arXiv

CORTEG: Foundation Models Enable Cross-Modality Representation Transfer from Scalp to Intracranial Brain Recordings

Intracranial electrocorticography (ECoG) offers high-signal-to-noise access to cortical activity for brain-computer interfaces, yet limited per-patient data has led most prior work to rely on small, subject-specific decoders that neglect information shared across patients. We investigate whether large pretrained scalp-EEG foundation models (EEG FMs) can be adapted to ECoG, enabling cross-patient learning and competitive decoding performance while calibrating to a held-out patient in 10-30 minutes on a single GPU. We introduce CORTEG, a cross-modality transfer framework that combines a pretrained EEG FM backbone, an electrode-aware KNNSoftFourier spatial adapter, a dual-stream tokenizer for low-frequency and high-gamma activity, and a leave-one-subject-out fine-tuning strategy. We evaluate CORTEG on two challenging regression tasks: public finger trajectory regression (n=9) and private audio envelope regression (n=16). CORTEG matches or exceeds the strongest task-specific baselines on both tasks: it reaches the highest mean correlation among compared methods on the public finger benchmark (gain not statistically significant on n=9 subjects), with larger and statistically significant gains on the audio task and in low-data per-patient calibration. Feature analyses align with neurophysiology, and latent manifolds capture low-dimensional finger-movement structure. CORTEG provides systematic evidence that scalp-EEG pretraining can be repurposed for ECoG decoding, enabling data-efficient intracranial BCIs that can adapt to new patients.

preprint2026arXiv

dFlowGRPO: Rate-Aware Policy Optimization for Discrete Flow Models

Discrete flow models (DFMs) are a class of flexible generative models for generating discrete data, and diffusion large language models (dLLMs) can be viewed as a special case with a specific choice of mixture path and a masked source distribution. While several recent works have explored reinforcement learning into dLLMs, its application to more general discrete flow models remains underexplored. In this work, we present discrete Flow-GRPO (dFlowGRPO), a unified reinforcement learning framework for discrete flow models that supports a broad family of probability paths and non-masked source distributions. We derive the full trajectory probability for DFMs and formulate denoising as a Markov decision process, enabling dFlowGRPO to incorporate information from both the associated conditional transition rates and the posterior model during reinforcement learning. We apply dFlowGRPO to FUDOKI, a recent multimodal discrete flow model, and evaluate it on both image generation and multimodal understanding tasks. Empirical results show that dFlowGRPO outperforms existing GRPO-type methods for dLLMs on text-to-image generation tasks and achieves performance competitive with continuous flow-based models trained using FlowGRPO, while also demonstrating strong capabilities on understanding tasks.

preprint2026arXiv

Docs2Synth: A Synthetic Data Trained Retriever Framework for Scanned Visually Rich Documents Understanding

Document understanding (VRDU) in regulated domains is particularly challenging, since scanned documents often contain sensitive, evolving, and domain specific knowledge. This leads to two major challenges: the lack of manual annotations for model adaptation and the difficulty for pretrained models to stay up-to-date with domain-specific facts. While Multimodal Large Language Models (MLLMs) show strong zero-shot abilities, they still suffer from hallucination and limited domain grounding. In contrast, discriminative Vision-Language Pre-trained Models (VLPMs) provide reliable grounding but require costly annotations to cover new domains. We introduce Docs2Synth, a synthetic-supervision framework that enables retrieval-guided inference for private and low-resource domains. Docs2Synth automatically processes raw document collections, generates and verifies diverse QA pairs via an agent-based system, and trains a lightweight visual retriever to extract domain-relevant evidence. During inference, the retriever collaborates with an MLLM through an iterative retrieval--generation loop, reducing hallucination and improving response consistency. We further deliver Docs2Synth as an easy-to-use Python package, enabling plug-and-play deployment across diverse real-world scenarios. Experiments on multiple VRDU benchmarks show that Docs2Synth substantially enhances grounding and domain generalization without requiring human annotations.

preprint2026arXiv

PRISMA: Reinforcement Learning Guided Two-Stage Policy Optimization in Multi-Agent Architecture for Open-Domain Multi-Hop Question Answering

Answering real-world open-domain multi-hop questions over massive corpora is a critical challenge in Retrieval-Augmented Generation (RAG) systems. Recent research employs reinforcement learning (RL) to end-to-end optimize the retrieval-augmented reasoning process, directly enhancing its capacity to resolve complex queries. However, reliable deployment is hindered by two obstacles. 1) Retrieval Collapse: iterative retrieval over large corpora fails to locate intermediate evidence containing bridge answers without reasoning-guided planning, causing downstream reasoning to collapse. 2) Learning Instability: end-to-end trajectory training suffers from weak credit assignment across reasoning chains and poor error localization across modules, causing overfitting to benchmark-specific heuristics that limit transferability and stability. To address these problems, we propose PRISMA, a decoupled RL-guided framework featuring a Plan-Retrieve-Inspect-Solve-Memoize architecture. PRISMA's strength lies in reasoning-guided collaboration: the Inspector provides reasoning-based feedback to refine the Planner's decomposition and fine-grained retrieval, while enforcing evidence-grounded reasoning in the Solver. We optimize individual agent capabilities via Two-Stage Group Relative Policy Optimization (GRPO). Stage I calibrates the Planner and Solver as specialized experts in planning and reasoning, while Stage II utilizes Observation-Aware Residual Policy Optimization (OARPO) to enhance the Inspector's ability to verify context and trigger targeted recovery. Experiments show that PRISMA achieves state-of-the-art performance on ten benchmarks and can be deployed efficiently in real-world scenarios.

preprint2025arXiv

Quantifying the advantage of vector over scalar magnetic sensor networks for undersea surveillance

Magnetic monitoring of maritime environments is an important problem for monitoring and optimising shipping, as well as national security. New developments in compact, fibre-coupled quantum magnetometers have led to the opportunity to critically evaluate how best to create such a sensor network. Here we explore various magnetic sensor network architectures for target identification. Our modelling compares networks of scalar vs vector magnetometers. We implement an unscented Kalman filter approach to perform target tracking, and we find that vector networks provide a significant improvement in target tracking, specifically tracking accuracy and resilience compared with scalar networks.

preprint2023arXiv

Online Linearized LASSO

Sparse regression has been a popular approach to perform variable selection and enhance the prediction accuracy and interpretability of the resulting statistical model. Existing approaches focus on offline regularized regression, while the online scenario has rarely been studied. In this paper, we propose a novel online sparse linear regression framework for analyzing streaming data when data points arrive sequentially. Our proposed method is memory efficient and requires less stringent restricted strong convexity assumptions. Theoretically, we show that with a properly chosen regularization parameter, the $\ell_2$-norm statistical error of our estimator diminishes to zero in the optimal order of $\tilde{O}({\sqrt{s/t}})$, where $s$ is the sparsity level, $t$ is the streaming sample size, and $\tilde{O}(\cdot)$ hides logarithmic terms. Numerical experiments demonstrate the practical efficiency of our algorithm.

preprint2022arXiv

A non-singular, field-only surface integral method for interactions between electric and magnetic dipoles and nano-structures

With the development of condensed-matter physics and nanotechnology, attention has turned to the fields near and on surfaces that result from interactions between electric dipole radiation and mesoscale structures. It is hoped that studying these fields will further our understanding of optical phenomena in nano-optics, quantum mechanics, electromagnetics and sensing using solid-state photon emitters. Here, we describe a method for implementing dynamic electric and magnetic dipoles in the frequency domain into a non-singular field-only surface integral method. We show that the effect of dipoles can conveniently be described as a relatively simple term in the integral equations, which fully represents how they drive the fields and interactions. Also, due to the non-singularity, our method can calculate the electric and magnetic fields on the surfaces of objects in both near and far fields with the same accuracy, which makes it an ideal tool to investigate nano-optical phenomena. The derivation of the framework is given and tested against a Mie theory alike formula. Some interesting examples are shown involving the interaction of dipoles with different types of mesoscale structures including parabolic nano-antennas and gold probes.

preprint2022arXiv

A universal model for drag on a spherical bubble

A theoretical expression for the drag on a spherical bubble is derived for the entire range from very viscous to inertial flow conditions. It is based on a solution for only that part of the velocity profile that determines the drag. It is assumed the surface of the bubble has a zero tangential stress condition. Excellent agreement with a previously proposed empirical model by Mei et al. is obtained. This shows that a theoretical framework with relatively simple physics can still predict the terminal velocity of a spherical bubble accurately. To the best of our knowledge, this is one of the few models in fluid dynamics to predict drag on an object for a range of Reynolds numbers that spans many orders of magnitude.

preprint2022arXiv

Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

The analytical solution is given for a vibrating rigid core sphere, oscillating up and down without volume change, situated at the center of an elastic material spherical shell, which in turn is situated inside an infinite (possible different) elastic medium. The solution is based on symmetry considerations and the continuity of the displacement both at the core and the shell - outer medium boundaries as well as the continuity of the stress at the outer edge of the shell. Furthermore, a separation into longitudinal and transverse waves is used. Analysis of the solution shows that a surprisingly complex range of physical phenomena can be observed when the frequency is changed while keeping the material parameters the same, especially when compared to the case of a core without any shell. With a careful choice of materials, shell thickness and vibration frequency, it is possible to filter out most of the longitudinal waves and generate pure tangential waves in the infinite domain (and vice-versa, we can filter out the tangential waves and generate longitudinal waves). When the solution is applied to different frequencies and with the help of a fast Fourier transform (FFT), a pulsed vibration is shown to exhibit the separation of the longitudinal (L) and transverse (T) waves (often called P- and S-waves in earthquake terminology).

preprint2022arXiv

Distributed Sparse Multicategory Discriminant Analysis

This paper proposes a convex formulation for sparse multicategory linear discriminant analysis and then extend it to the distributed setting when data are stored across multiple sites. The key observation is that for the purpose of classification it suffices to recover the discriminant subspace which is invariant to orthogonal transformations. Theoretically, we establish statistical properties ensuring that the distributed sparse multicategory linear discriminant analysis performs as good as the centralized version after {a few rounds} of communications. Numerical studies lend strong support to our methodology and theory.

preprint2022arXiv

Efficient sunlight promoted nitrogen fixation from air under room temperature and ambient pressure via Ti/Mo composites

Photocatalytic nitrogen fixation is an important pathway for carbon neutralization and sustainable development. Inspired by nitrogenase, the participation of molybdenum can effectively activate nitrogen. A novel Ti/Mo composites photocatalyst is designed by sintering the molybdenum acetylacetonate precursor with TiO$_{2}$. The special carbon-coated hexagonal photocatalyst is obtained which photocatalytic nitrogen fixation performance is enhanced 16 times compared to pure TiO$_{2}$ at room temperature and ambient pressure. The abundant surface defects in this composite were confirmed to be the key factor for nitrogen fixation. The $^{15}$N$_{2}$ isotope labeling experiment was used to demonstrate the feasibility of nitrogen to ammonia conversion. Also, modelling on the interactions between light and the synthesized photocatalyst particle was examined for the light absorption. The optimum nitrogen fixation conditions have been examined, and the nitrogen fixation performance can reach up to 432 $μ$g$\cdot$g$_{\text{cat}}^{-1}\cdot$h$^{-1}$. Numerical simulations via the field-only surface integral method were also carried out to study the interactions between light and the photocatalytic particles to further confirm that it can be a useful material for photocatalyst. This newly developed Ti/Mo composites provide a simple and effective strategy for photocatalytic nitrogen fixation from air directly under ambient conditions.

preprint2022arXiv

Growth optimization and device integration of narrow-bandgap graphene nanoribbons

The electronic, optical and magnetic properties of graphene nanoribbons (GNRs) can be engineered by controlling their edge structure and width with atomic precision through bottom-up fabrication based on molecular precursors. This approach offers a unique platform for all-carbon electronic devices but requires careful optimization of the growth conditions to match structural requirements for successful device integration, with GNR length being the most critical parameter. In this work, we study the growth, characterization, and device integration of 5-atom wide armchair GNRs (5-AGNRs), which are expected to have an optimal band gap as active material in switching devices. 5-AGNRs are obtained via on-surface synthesis under ultra-high vacuum conditions from Br- and I-substituted precursors. We show that the use of I-substituted precursors and the optimization of the initial precursor coverage quintupled the average 5-AGNR length. This significant length increase allowed us to integrate 5-AGNRs into devices and to realize the first field-effect transistor based on narrow bandgap AGNRs that shows switching behavior at room temperature. Our study highlights that optimized growth protocols can successfully bridge between the sub-nanometer scale, where atomic precision is needed to control the electronic properties, and the scale of tens of nanometers relevant for successful device integration of GNRs.

preprint2022arXiv

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time from a theoretical point of view, the actual numerical implementation has often been hindered by divergence free and/or curl free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method (BEM) satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional (3D) boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.

preprint2022arXiv

Length-independent quantum transport through topological band states of graphene nanoribbons

Atomically precise graphene nanoribbons (GNRs) have emerged as promising candidates for nanoelectronic applications due to their widely tunable energy band gaps resulting from lateral quantum confinement and edge effects. Here we report on the electronic transport characterization of an edge-modified GNR suspended between the tip of a scanning tunneling microscope (STM) and a Au(111) substrate. Differential conductance measurements on this metal-GNR-metal junction reveal loss-less transport properties (inverse decay length $β< 0.001 /\overset{\circ}{\mathrm{A}}$) with high conductance ($\sim 0.1$ G$_0$) at low voltages (50 meV) over long distances ($z > 10$ nm). The transport behavior is sensitive to the coupling between ribbon and electrodes, an effect that is rationalized using tight-binding and density functional theory simulations. From extensive modelling we infer that the length-independent transport is a manifestation of band transport through topological valence states, which originate from the zigzag segments on the GNR edges.

preprint2022arXiv

Local Slot Attention for Vision-and-Language Navigation

Vision-and-language navigation (VLN), a frontier study aiming to pave the way for general-purpose robots, has been a hot topic in the computer vision and natural language processing community. The VLN task requires an agent to navigate to a goal location following natural language instructions in unfamiliar environments. Recently, transformer-based models have gained significant improvements on the VLN task. Since the attention mechanism in the transformer architecture can better integrate inter- and intra-modal information of vision and language. However, there exist two problems in current transformer-based models. 1) The models process each view independently without taking the integrity of the objects into account. 2) During the self-attention operation in the visual modality, the views that are spatially distant can be inter-weaved with each other without explicit restriction. This kind of mixing may introduce extra noise instead of useful information. To address these issues, we propose 1) A slot-attention based module to incorporate information from segmentation of the same object. 2) A local attention mask mechanism to limit the visual attention span. The proposed modules can be easily plugged into any VLN architecture and we use the Recurrent VLN-Bert as our base model. Experiments on the R2R dataset show that our model has achieved the state-of-the-art results.

preprint2022arXiv

Modified Multidimensional Scaling and High Dimensional Clustering

Multidimensional scaling is an important dimension reduction tool in statistics and machine learning. Yet few theoretical results characterizing its statistical performance exist, not to mention any in high dimensions. By considering a unified framework that includes low, moderate and high dimensions, we study multidimensional scaling in the setting of clustering noisy data. Our results suggest that, the classical multidimensional scaling can be modified to further improve the quality of embedded samples, especially when the noise level increases. To this end, we propose {\it modified multidimensional scaling} which applies a nonlinear transformation to the sample eigenvalues. The nonlinear transformation depends on the dimensionality, sample size and moment of noise. We show that modified multidimensional scaling followed by various clustering algorithms can achieve exact recovery, i.e., all the cluster labels can be recovered correctly with probability tending to one. Numerical simulations and two real data applications lend strong support to our proposed methodology.

preprint2020arXiv

Bayesian high-dimensional linear regression with generic spike-and-slab priors

Spike-and-slab priors are popular Bayesian solutions for high-dimensional linear regression problems. Previous theoretical studies on spike-and-slab methods focus on specific prior formulations and use prior-dependent conditions and analyses, and thus can not be generalized directly. In this paper, we propose a class of generic spike-and-slab priors and develop a unified framework to rigorously assess their theoretical properties. Technically, we provide general conditions under which generic spike-and-slab priors can achieve the nearly-optimal posterior contraction rate and the model selection consistency. Our results include those of Narisetty and He (2014) and Castillo et al. (2015) as special cases.

preprint2020arXiv

Exact Cluster Recovery via Classical Multidimensional Scaling

Classical multidimensional scaling is an important dimension reduction technique. Yet few theoretical results characterizing its statistical performance exist. This paper provides a theoretical framework for analyzing the quality of embedded samples produced by classical multidimensional scaling. This lays the foundation for various downstream statistical analyses, and we focus on clustering noisy data. Our results provide scaling conditions on the sample size, ambient dimensionality, between-class distance, and noise level under which classical multidimensional scaling followed by a distance-based clustering algorithm can recover the cluster labels of all samples with high probability. Numerical simulations confirm these scaling conditions are near-sharp. Applications to both human genomics data and natural language data lend strong support to the methodology and theory.

preprint2020arXiv

Iteratively Reweighted $\ell_1$-Penalized Robust Regression

This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear models have only bounded second moment, we show that iteratively reweighted $\ell_1$-penalized adaptive Huber regression estimator satisfies exponential deviation bounds and oracle properties, including the oracle convergence rate and variable selection consistency, under a weak beta-min condition. Computationally, we need as many as $O(\log s + \log\log d)$ iterations to reach such an oracle estimator, where $s$ and $d$ denote the sparsity and ambient dimension, respectively. Extension to a general class of robust loss functions is also considered. Numerical studies lend strong support to our methodology and theory.

preprint2020arXiv

Learning to Augment Expressions for Few-shot Fine-grained Facial Expression Recognition

Affective computing and cognitive theory are widely used in modern human-computer interaction scenarios. Human faces, as the most prominent and easily accessible features, have attracted great attention from researchers. Since humans have rich emotions and developed musculature, there exist a lot of fine-grained expressions in real-world applications. However, it is extremely time-consuming to collect and annotate a large number of facial images, of which may even require psychologists to correctly categorize them. To the best of our knowledge, the existing expression datasets are only limited to several basic facial expressions, which are not sufficient to support our ambitions in developing successful human-computer interaction systems. To this end, a novel Fine-grained Facial Expression Database - F2ED is contributed in this paper, and it includes more than 200k images with 54 facial expressions from 119 persons. Considering the phenomenon of uneven data distribution and lack of samples is common in real-world scenarios, we further evaluate several tasks of few-shot expression learning by virtue of our F2ED, which are to recognize the facial expressions given only few training instances. These tasks mimic human performance to learn robust and general representation from few examples. To address such few-shot tasks, we propose a unified task-driven framework - Compositional Generative Adversarial Network (Comp-GAN) learning to synthesize facial images and thus augmenting the instances of few-shot expression classes. Extensive experiments are conducted on F2ED and existing facial expression datasets, i.e., JAFFE and FER2013, to validate the efficacy of our F2ED in pre-training facial expression recognition network and the effectiveness of our proposed approach Comp-GAN to improve the performance of few-shot recognition tasks.

preprint2020arXiv

Question Guided Modular Routing Networks for Visual Question Answering

This paper studies the task of Visual Question Answering (VQA), which is topical in Multimedia community recently. Particularly, we explore two critical research problems existed in VQA: (1) efficiently fusing the visual and textual modalities; (2) enabling the visual reasoning ability of VQA models in answering complex questions. To address these challenging problems, a novel Question Guided Modular Routing Networks (QGMRN) has been proposed in this paper. Particularly, The QGMRN is composed of visual, textual and routing network. The visual and textual network serve as the backbones for the generic feature extractors of visual and textual modalities. QGMRN can fuse the visual and textual modalities at multiple semantic levels. Typically, the visual reasoning is facilitated by the routing network in a discrete and stochastic way by using Gumbel-Softmax trick for module selection. When the input reaches a certain modular layer, routing network newly proposed in this paper, dynamically selects a portion of modules from that layer to process the input depending on the question features generated by the textual network. It can also learn to reason by routing between the generic modules without additional supervision information or expert knowledge. Benefiting from the dynamic routing mechanism, QGMRN can outperform the previous classical VQA methods by a large margin and achieve the competitive results against the state-of-the-art methods. Furthermore, attention mechanism is integrated into our QGMRN model and thus can further boost the model performance. Empirically, extensive experiments on the CLEVR and CLEVR-Humans datasets validate the effectiveness of our proposed model, and the state-of-the-art performance has been achieved.

preprint2019arXiv

On-surface synthesis of polyazulene with 2,6-connectivity

Azulene, the smallest neutral nonalternant aromatic hydrocarbon, serves as not only a prototype for fundamental studies but also a versatile building block for functional materials because of its unique opto(electronic) properties. Here, we report the on-surface synthesis and characterization of the homopolymer of azulene connected exclusively at the 2,6-positions using 2,6-diiodoazulene as the monomer precursor. As an intermediate to the formation of polyazulene, a gold-(2,6-azulenylene) chain is observed.

preprint2019arXiv

Robust Field-Only Surface Integral Equations: Scattering from a Dielectric Body

A robust and efficient field-only nonsingular surface integral method to solve Maxwell&#39;s equations for the components of the electric field on the surface of a dielectric scatterer is introduced. In this method, both the vector Helmholtz equation and the divergence-free constraint are satisfied inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric field across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic fields are expressed in terms of the electric field across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric field inside and outside the scatterer is solved by a fully desingularized surface integral method. Comparing with the most popular surface integral methods based on the Stratton-Chu formulation or the PMCHWT formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents and the use of complicated vector basis functions can be avoided altogether. Also, our method is not affected by numerical issues such as the zero frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric field and the magnetic field, our theoretical framework can also be used to solve for the magnetic field.

preprint2019arXiv

Robust Field-Only Surface Integral Equations: Scattering from a Perfect Electric Conductor

A robust field-only boundary integral formulation of electromagnetics is derived without the use of surface currents that appear in the Stratton-Chu formulation. For scattering by a perfect electrical conductor (PEC), the components of the electric field are obtained directly from surface integral equation solutions of three scalar Helmholtz equations for the field components. The divergence-free condition is enforced via a boundary condition on the normal component of the field and its normal derivative. Field values and their normal derivatives at the surface of the PEC are obtained directly from surface integral equations that do not contain divergent kernels. Consequently, high-order elements with fewer degrees of freedom can be used to represent surface features to a higher precision than the traditional planar elements. This theoretical framework is illustrated with numerical examples that provide further physical insight into the role of the surface curvature in scattering problems.

preprint2017arXiv

Locating any two vertices on Hamiltonian cycles

In this paper we give a proof of Enomoto&#39;s conjecture for graphs of sufficiently large order. Enomoto&#39;s conjecture states that, if $G$ is a graph of order $n$ with minimum degree $δ(G)\geq \frac{n}{2}+1$, then for any pair of vertices $x$, $y$ in $G$, there is a Hamiltonian cycle $C$ of $G$ such that $d_C(x,y)=\lfloor \frac{n}{2}\rfloor$. The main tools of our proof are Regularity Lemma of Szemerédi and Blow-up Lemma of Komlós et al.

preprint2015arXiv

Walk-powers and homomorphism bound of planar graphs

As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least $2k+1$ admits a homomorphism to $PC_{2k}=(\mathbb{Z}_2^{2k}, \{e_1, e_2, ...,e_{2k}, J\})$ where $e_i$&#39;s are standard basis and $J$ is all 1 vector. Noting that $PC_{2k}$ itself is of odd-girth $2k+1$, in this work we show that if the conjecture is true, then $PC_{2k}$ is an optimal such a graph both with respect to number of vertices and number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for bipartite signed planar graphs of unbalanced-girth $2k$. The work is presented on a uniform frame work of planar consistent signed graphs.