arXiv subjects

Aditya Siripuram

Publications and source records attributed to Aditya Siripuram.

13 recordsLinked to original sources

arXiv2025

Exploring the Limitations of Structured Orthogonal Dictionary Learning

This work is motivated by recent applications of structured dictionary learning, in particular when the dictionary is assumed to be the product of a few Householder atoms. We investigate the following two problems: 1) How do we approximate an orthogonal matrix $\mathbf{V}$ with a product of a specified number of Householder matrices, and 2) How many samples are required to learn a structured (Householder) dictionary from data? For 1) we discuss an algorithm that decomposes $\mathbf{V}$ as a product of a specified number of Householder matrices. We see that the algorithm outputs the decomposition when it exists, and give bounds on the approximation error of the algorithm when such a decomposition does not exist. For 2) given data $\mathbf{Y}=\mathbf{HX}$, we show that when assuming a binary coefficient matrix $\mathbf{X}$, the structured (Householder) dictionary learning problem can be solved with just $2$ samples (columns) in $\mathbf{Y}$.

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arXiv2025

Fast Structured Orthogonal Dictionary Learning using Householder Reflections

In this paper, we propose and investigate algorithms for the structured orthogonal dictionary learning problem. First, we investigate the case when the dictionary is a Householder matrix. We give sample complexity results and show theoretically guaranteed approximate recovery (in the $l_{\infty}$ sense) with optimal computational complexity. We then attempt to generalize these techniques when the dictionary is a product of a few Householder matrices. We numerically validate these techniques in the sample-limited setting to show performance similar to or better than existing techniques while having much improved computational complexity.

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arXiv2024

Efficient Matrix Factorization Via Householder Reflections

Motivated by orthogonal dictionary learning problems, we propose a novel method for matrix factorization, where the data matrix $\mathbf{Y}$ is a product of a Householder matrix $\mathbf{H}$ and a binary matrix $\mathbf{X}$. First, we show that the exact recovery of the factors $\mathbf{H}$ and $\mathbf{X}$ from $\mathbf{Y}$ is guaranteed with $Ω(1)$ columns in $\mathbf{Y}$ . Next, we show approximate recovery (in the $l\infty$ sense) can be done in polynomial time($O(np)$) with $Ω(\log n)$ columns in $\mathbf{Y}$ . We hope the techniques in this work help in developing alternate algorithms for orthogonal dictionary learning.

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arXiv2024

Joint Data Inpainting and Graph Learning via Unrolled Neural Networks

Given partial measurements of a time-varying graph signal, we propose an algorithm to simultaneously estimate both the underlying graph topology and the missing measurements. The proposed algorithm operates by training an interpretable neural network, designed from the unrolling framework. The proposed technique can be used both as a graph learning and a graph signal reconstruction algorithm. This work enhances prior work in graph signal reconstruction by allowing the underlying graph to be unknown; and also builds on prior work in graph learning by tailoring the learned graph to the signal reconstruction task.

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arXiv2024

Numerical Stability of DFT Computation for Signals with Structured Support

We consider the problem of building numerically stable algorithms for computing Discrete Fourier Transform (DFT) of $N$- length signals with known frequency support of size $k$. A typical algorithm, in this case, would involve solving (possibly poorly conditioned) system of equations, causing numerical instability. When $N$ is a power of 2, and the frequency support is a random subset of $\mathbb{Z}_N$, we provide an algorithm that has (a possibly optimal) $O(k \log k)$ complexity to compute the DFT while solving system of equations that are $O(1)$ in size.

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arXiv2022

Fast DFT Computation for Signals with Structured Support

Suppose an $N-$length signal has known frequency support of size $k$. Given sample access to this signal, how fast can we compute the DFT? The answer to this question depends on the structure of the frequency support. We first identify some frequency supports for which (an ideal) $O(k \log k)$ complexity is achievable, referred to as homogeneous sets. We give a generalization of radix-2 that enables $O(k\log k)$ computation of signals with homogeneous frequency support. Using homogeneous sets as building blocks, we construct more complicated support structures for which the complexity of $O(k\log k)$ is achievable. We also investigate the relationship of DFT computation with additive structure in the support and provide partial converses.

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arXiv2022

Investigating the relationship between graph eigenvector ordering and the signal processing dual

Graph signal processing uses the graph eigenvector basis to analyze signals. However, these graph eigenvectors are typically linearly ordered (by total variation), which may not be reasonable for many graph structures. There have been structure based similarity metrics proposed in the literature that better capture the geometry of graph eigenvectors. On the other hand, there has been work that attempts to generalize the concept of duality to graph signal processing. The (signal processing) the dual graph captures the relationship between graph frequencies and is obtained by typically inverting the graph Fourier transform operation. In this work, we investigate the connections between these two concepts. We propose a dualness measure of two graphs, which quantifies how close the graphs are being (signal processing) duals of each other. We show that this definition satisfies some desirable properties and develop an algorithm based on coordinate descent and perfect matching to compute an approximation to dualness. By computing the dualness measure, we observe that for structured graphs, the similarity metric-based techniques give a better dual graph (and vice-versa for ER graphs), suggesting a potentially novel approach unifying the two methods.

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arXiv2021

Can Perceptual Guidance Lead to Semantically Explainable Adversarial Perturbations?

It is well known that carefully crafted imperceptible perturbations can cause state-of-the-art deep learning classification models to misclassify. Understanding and analyzing these adversarial perturbations play a crucial role in the design of robust convolutional neural networks. However, their mechanics are not well understood. In this work, we attempt to understand the mechanics by systematically answering the following question: do imperceptible adversarial perturbations focus on changing the regions of the image that are important for classification? In other words, are imperceptible adversarial perturbations semantically explainable? Most current methods use $l_p$ distance to generate and characterize the imperceptibility of the adversarial perturbations. However, since $l_p$ distances only measure the pixel to pixel distances and do not consider the structure in the image, these methods do not provide a satisfactory answer to the above question. To address this issue, we propose a novel framework for generating adversarial perturbations by explicitly incorporating a "perceptual quality ball" constraint in our formulation. Specifically, we pose the adversarial example generation problem as a tractable convex optimization problem, with constraints taken from a mathematically amenable variant of the popular SSIM index. We use the MobileNetV2 network trained on the ImageNet dataset for our experiments. By comparing the SSIM maps generated by our method with class activation maps, we show that the perceptually guided perturbations introduce changes specifically in the regions that contribute to classification decisions i.e., these perturbations are semantically explainable.

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arXiv2021

Computing the Discrete Fourier Transform of signals with spectral frequency support

We consider the problem of finding the Discrete Fourier Transform (DFT) of $N-$ length signals with known frequency support of size $k$. When $N$ is a power of 2 and the frequency support is a spectral set, we provide an $O(k \log k)$ algorithm to compute the DFT. Our algorithm uses some recent characterizations of spectral sets and is a generalization of the standard radix-2 algorithm.

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arXiv2020

Graph learning under spectral sparsity constraints

Graph inference plays an essential role in machine learning, pattern recognition, and classification. Signal processing based approaches in literature generally assume some variational property of the observed data on the graph. We make a case for inferring graphs on which the observed data has high variation. We propose a signal processing based inference model that allows for wideband frequency variation in the data and propose an algorithm for graph inference. The proposed inference algorithm consists of two steps: 1) learning orthogonal eigenvectors of a graph from the data; 2) recovering the adjacency matrix of the graph topology from the given graph eigenvectors. The first step is solved by an iterative algorithm with a closed-form solution. In the second step, the adjacency matrix is inferred from the eigenvectors by solving a convex optimization problem. Numerical results on synthetic data show the proposed inference algorithm can effectively capture the meaningful graph topology from observed data under the wideband assumption.

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arXiv2020

Convolution Idempotents with a given Zero-set

We investigate the structure of N-length discrete signals h satisfying h*h=h that vanish on a given set of indices. We motivate this problem from examples in sampling, Fuglede's conjecture, and orthogonal interpolation of bandlimited signals. When N is a prime power, we characterize all such h with a prescribed zero set in terms of digit expansions of nonzero indices in the inverse DFT of h.

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arXiv2018

Discrete Sampling: A graph theoretic approach to Orthogonal Interpolation

We study the problem of finding unitary submatrices of the $N \times N$ discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on $\mathbb{Z}_N$ and tiling $\mathbb{Z}_N$. In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when $N$ is a prime power, and we identify the challenges in generalizing to arbitrary $N$. Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture.

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arXiv2012

Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces

We study the problem of interpolating all values of a discrete signal f of length N when d<N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J; these comprise the (generalized) bandlimited spaces B^J. The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B^J for any J. When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.

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