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.

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.

Perceptually Guided 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, the reasons for the existence of adversarial perturbations and their mechanics are not well understood. In this work, we attempt to systematically answer the following question: do imperceptible adversarial perturbations focus on changing the regions of the image that are important for classification? 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 show that the perturbations generated by the proposed method result in a high fooling rate with minimal impact on perceptual quality compared to the norm bounded adversarial perturbations. Further, through SSIM maps, we show that the perceptually guided perturbations introduce changes specifically in the regions that contribute to classification decisions. We use networks trained on MNIST and CIFAR-10 datasets quantitative analysis, and MobileNetV2 trained on the ImageNet dataset for further qualitative analysis.

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.