arXiv subjects

Sourya Roy

Publications and source records attributed to Sourya Roy.

15 recordsLinked to original sources

arXiv2026

Low Soundness Linearity Testing on the Half-Slice

Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$ such that $x,y,x+y\in T$, then $f$ agrees with some linear function on at least $\frac{1+\delta}{2}-o(1)$ fraction of the points in $T$. More generally, we show that if $f$ passes the natural $k$-query BLR test with probability $\frac{1+\delta}{2}$ for any $k\geq3$, then it must agree with some affine function at $\frac{1+\delta^{\frac{1}{k-2}}}{2}-o(1)$ fraction of the points in $T$. The only other known linearity test for the slice in the low soundness regime (i.e., when $\delta$ can be arbitrarily small) was given by Kalai, Lifshitz, Minzer, and Ziegler [FOCS'24]. Our result improves upon this result in two significant ways: firstly, it works for $k=3$ queries, instead of requiring $k\geq4$; secondly, our result is sharper, e.g., when $k=4$, we are able to conclude an agreement of $\frac{1+\sqrt{\delta}}{2}-o(1)$ instead of $\frac{1+c\sqrt{\delta}}{2}$ for $c\approx.0035$. In particular, our result matches (up to the $o(1)$ term) the conclusion one obtains over the full hypercube via the classical BLR analysis. Our main technical contribution is a new dense model theorem using bounds on Krawtchouk polynomials. Using these Krawtchouk polynomial bounds, we also obtain a simple $k$-query test ($k\geq 5$) that avoids any use of the dense model machinery. This simplified test naturally extends to the slice over the $q$-ary hypercube, giving the first such result over larger alphabets.

cs.CC↗

arXiv2025

A General Framework for Low Soundness Homomorphism Testing

We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and $\mathbb{Z}_p$, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of $\mathrm{GL}_n(\mathbb{F}_q)$, and finite-dimensional Lie algebras over $\mathbb{F}_q$. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between $\mathbb{F}_q^n$ and $\mathbb{F}_q$. Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as $\mathbb{F}_q^n$). No tests were known for non-abelian groups. As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of $O(\varepsilon^{-2})$ (for agreement parameter $\varepsilon$). This improves upon the currently best-known bound of $O(\varepsilon^{-105})$ due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

cs.CC↗

arXiv2025

Pseudorandomness of Expander Walks via Fourier Analysis on Groups

One approach to study the pseudorandomness properties of walks on expander graphs is to label the vertices of an expander with elements from an alphabet $\Sigma$, and study the mean of functions over $\Sigma^n$. We say expander walks $\varepsilon$-fool a function if, for any unbiased labeling of the vertices, the expander walk mean is $\varepsilon$-close to the true mean. We show that: - The class of symmetric functions is $O(|\Sigma|\cdot\lambda)$-fooled by expander walks over any generic $\lambda$-expander, and any alphabet $\Sigma$ . This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for $|\Sigma| =2$, and exponentially improves the previous bound of $O(|\Sigma|^{O(|\Sigma|)}\cdot \lambda)$, by Golowich and Vadhan [CCC'22]. Additionally, if the expander is a Cayley graph over $\mathbb{Z}_{|\Sigma|}$, we get a further improved bound of $O(\sqrt{|\Sigma|}\cdot\lambda)$. Morever, when $\Sigma$ is a finite group $G$, we show the following for functions over $G^n$: - The class of symmetric class functions is $O\Big({\frac{\sqrt{|G|}}{D}\cdot\lambda}\Big)$-fooled by expander walks over "structured" $\lambda$-expanders, if $G$ is $D$-quasirandom. - We show a lower bound of $\Omega(\lambda)$ for symmetric functions for any finite group $G$ (even for "structured" $\lambda$-expanders). - We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks. Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of $\mathbb{Z}_2$ or $\mathbb{Z}$. This enables us to get quantitatively better bounds even for unstructured sets.

cs.CC↗

arXiv2024

Sublinear-Time Sampling of Spanning Trees in the Congested Clique

We present the first sublinear-in-$n$ round algorithm for sampling an approximately uniform spanning tree of an $n$-vertex graph in the CongestedClique model of distributed computing. In particular, our algorithm requires $\Tilde{O}(n^{0.657})$ rounds for sampling a spanning tree within total variation distance $1/n^c$, for arbitrary constant $c > 0$, from the uniform distribution. More precisely, our algorithm requires $\Tilde{O}(n^{1/2 + \alpha})$ rounds, where $O(n^\alpha)$ is the running time of matrix multiplication in the CongestedClique model (currently $\alpha = 1 - 2/\omega = 0.157$, where $\omega$ is the sequential matrix multiplication time exponent). We can adapt our algorithm to give exact rather than approximate samples, but with a larger, though still $o(n)$, runtime of $\Tilde{O}(n^{2/3+\alpha}) = O(n^{.824})$. In a remarkable result, Aldous (SIDM 1990) and Broder (FOCS 1989) showed that the first visit edge to each vertex, excluding the start vertex, during a random walk forms a uniformly chosen spanning tree of the underlying graph. Our algorithm is a significant departure from known techniques, featuring a top-down walk filling approach paired with Schur complement graphs for walk shortcutting. To make this idea work in the CongestedClique model, we present a novel compressed random walk reconstruction algorithm, based on randomly sampling a weighted perfect matching. In addition, we show how to take somewhat shorter random walks even more efficiently in the CongestedClique model, obtaining an $O(\log^3 n)$-round algorithm for uniformly sampling spanning trees from graphs with $O(n\log n)$ cover times. These results are obtained by adding a load balancing component to the random walk algorithm of Bahmani, Chakrabarti and Xin (SIGMOD 2011) that uses the bottom-up ``doubling'' technique.

cs.DC↗

arXiv2024

Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime

We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08] and Gowers [Comb. Probab. Comput.'08].

cs.CC↗

arXiv2022

Learning Long-Term Spatial-Temporal Graphs for Active Speaker Detection

Active speaker detection (ASD) in videos with multiple speakers is a challenging task as it requires learning effective audiovisual features and spatial-temporal correlations over long temporal windows. In this paper, we present SPELL, a novel spatial-temporal graph learning framework that can solve complex tasks such as ASD. To this end, each person in a video frame is first encoded in a unique node for that frame. Nodes corresponding to a single person across frames are connected to encode their temporal dynamics. Nodes within a frame are also connected to encode inter-person relationships. Thus, SPELL reduces ASD to a node classification task. Importantly, SPELL is able to reason over long temporal contexts for all nodes without relying on computationally expensive fully connected graph neural networks. Through extensive experiments on the AVA-ActiveSpeaker dataset, we demonstrate that learning graph-based representations can significantly improve the active speaker detection performance owing to its explicit spatial and temporal structure. SPELL outperforms all previous state-of-the-art approaches while requiring significantly lower memory and computational resources. Our code is publicly available at https://github.com/SRA2/SPELL

cs.CV↗

arXiv2022

Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/\lambda^{2+o(1)}$) trade-off between (any desired) spectral expansion $\lambda$ and degree $d$. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius $O(\log(1/\lambda))$. The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert--Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.

cs.DS↗

arXiv2022

Analyzing Ta-Shma's Code via the Expander Mixing Lemma

Random walks in expander graphs and their various derandomizations (e.g., replacement/zigzag product) are invaluable tools from pseudorandomness. Recently, Ta-Shma used s-wide replacement walks in his breakthrough construction of a binary linear code almost matching the Gilbert-Varshamov bound (STOC 2017). Ta-Shma's original analysis was entirely linear algebraic, and subsequent developments have inherited this viewpoint. In this work, we rederive Ta-Shma's analysis from a combinatorial point of view using repeated application of the expander mixing lemma. We hope that this alternate perspective will yield a better understanding of Ta-Shma's construction. As an additional application of our techniques, we give an alternate proof of the expander hitting set lemma.

cs.IT↗

arXiv2021

Learning Spatial-Temporal Graphs for Active Speaker Detection

We address the problem of active speaker detection through a new framework, called SPELL, that learns long-range multimodal graphs to encode the inter-modal relationship between audio and visual data. We cast active speaker detection as a node classification task that is aware of longer-term dependencies. We first construct a graph from a video so that each node corresponds to one person. Nodes representing the same identity share edges between them within a defined temporal window. Nodes within the same video frame are also connected to encode inter-person interactions. Through extensive experiments on the Ava-ActiveSpeaker dataset, we demonstrate that learning graph-based representation, owing to its explicit spatial and temporal structure, significantly improves the overall performance. SPELL outperforms several relevant baselines and performs at par with state of the art models while requiring an order of magnitude lower computation cost.

cs.CV↗

arXiv2021

Mixing of 3-term progressions in Quasirandom Groups

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom groups, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1, A_2, A_3 \subset G$, we have \[ \left|\Pr_{x,y\sim G}\left[ x \in A_1, xy \in A_2, xy^2 \in A_3\right] - \prod_{i=1}^3 \Pr_{x\sim G}\left[x \in A_i\right] \right| \leq \left(\frac{2}{\sqrt{D}}\right)^{\frac{1}{4}}.\] Prior to this, Tao answered this question when the underlying quasirandom group is $\mathrm{SL}_{d}(\mathbb{F}_q)$. Subsequently, Peluse extended the result to all nonabelian finite $\textit{simple}$ groups. In this work, we show that a slight modification of Peluse's argument is sufficient to fully resolve Gower's quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs of Tao and Peluse, our proof is elementary and only uses basic facts from nonabelian Fourier analysis.

math.CO↗

arXiv2018

W-TALC: Weakly-supervised Temporal Activity Localization and Classification

Most activity localization methods in the literature suffer from the burden of frame-wise annotation requirement. Learning from weak labels may be a potential solution towards reducing such manual labeling effort. Recent years have witnessed a substantial influx of tagged videos on the Internet, which can serve as a rich source of weakly-supervised training data. Specifically, the correlations between videos with similar tags can be utilized to temporally localize the activities. Towards this goal, we present W-TALC, a Weakly-supervised Temporal Activity Localization and Classification framework using only video-level labels. The proposed network can be divided into two sub-networks, namely the Two-Stream based feature extractor network and a weakly-supervised module, which we learn by optimizing two complimentary loss functions. Qualitative and quantitative results on two challenging datasets - Thumos14 and ActivityNet1.2, demonstrate that the proposed method is able to detect activities at a fine granularity and achieve better performance than current state-of-the-art methods.

cs.CV↗

arXiv2018

Incorporating Scalability in Unsupervised Spatio-Temporal Feature Learning

Deep neural networks are efficient learning machines which leverage upon a large amount of manually labeled data for learning discriminative features. However, acquiring substantial amount of supervised data, especially for videos can be a tedious job across various computer vision tasks. This necessitates learning of visual features from videos in an unsupervised setting. In this paper, we propose a computationally simple, yet effective, framework to learn spatio-temporal feature embedding from unlabeled videos. We train a Convolutional 3D Siamese network using positive and negative pairs mined from videos under certain probabilistic assumptions. Experimental results on three datasets demonstrate that our proposed framework is able to learn weights which can be used for same as well as cross dataset and tasks.

cs.CV↗

arXiv2016

WEPSAM: Weakly Pre-Learnt Saliency Model

Visual saliency detection tries to mimic human vision psychology which concentrates on sparse, important areas in natural image. Saliency prediction research has been traditionally based on low level features such as contrast, edge, etc. Recent thrust in saliency prediction research is to learn high level semantics using ground truth eye fixation datasets. In this paper we present, WEPSAM : Weakly Pre-Learnt Saliency Model as a pioneering effort of using domain specific pre-learing on ImageNet for saliency prediction using a light weight CNN architecture. The paper proposes a two step hierarchical learning, in which the first step is to develop a framework for weakly pre-training on a large scale dataset such as ImageNet which is void of human eye fixation maps. The second step refines the pre-trained model on a limited set of ground truth fixations. Analysis of loss on iSUN and SALICON datasets reveal that pre-trained network converges much faster compared to randomly initialized network. WEPSAM also outperforms some recent state-of-the-art saliency prediction models on the challenging MIT300 dataset.

cs.CV↗

arXiv2016

Visual saliency detection: a Kalman filter based approach

In this paper we propose a Kalman filter aided saliency detection model which is based on the conjecture that salient regions are considerably different from our "visual expectation" or they are "visually surprising" in nature. In this work, we have structured our model with an immediate objective to predict saliency in static images. However, the proposed model can be easily extended for space-time saliency prediction. Our approach was evaluated using two publicly available benchmark data sets and results have been compared with other existing saliency models. The results clearly illustrate the superior performance of the proposed model over other approaches.

cs.CV↗

arXiv2011

Elastic Fidelity: Trading-off Computational Accuracy for Energy Reduction

Power dissipation and energy consumption have become one of the most important problems in the design of processors today. This is especially true in power-constrained environments, such as embedded and mobile computing. While lowering the operational voltage can reduce power consumption, there are limits imposed at design time, beyond which hardware components experience faulty operation. Moreover, the decrease in feature size has led to higher susceptibility to process variations, leading to reliability issues and lowering yield. However, not all computations and all data in a workload need to maintain 100% fidelity. In this paper, we explore the idea of employing functional or storage units that let go the conservative guardbands imposed on the design to guarantee reliable execution. Rather, these units exhibit Elastic Fidelity, by judiciously lowering the voltage to trade-off reliable execution for power consumption based on the error guarantees required by the executing code. By estimating the accuracy required by each computational segment of a workload, and steering each computation to different functional and storage units, Elastic Fidelity Computing obtains power and energy savings while reaching the reliability targets required by each computational segment. Our preliminary results indicate that even with conservative estimates, Elastic Fidelity can reduce the power and energy consumption of a processor by 11-13% when executing applications involving human perception that are typically included in modern mobile platforms, such as audio, image, and video decoding.

cs.AR↗