Deterministic Sparse Fourier Transform with an ?_{?} Guarantee

In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x \in \mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $\hat x$, the Discrete Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.\@ (J Fourier Anal Appl 2018), which obtains $O(k^2 \log^{-1}k \cdot \log^{5.5}n)$ samples and a similar runtime with the $\ell_2/\ell_1$ guarantee. We focus on the stronger $\ell_{\infty}/\ell_1$ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We find a deterministic collection of $O(k^2 \log n)$ samples for the $\ell_\infty/\ell_1$ recovery in time $O(nk \log^2 n)$, and a deterministic collection of $O(k^2 \log^2 n)$ samples for the $\ell_\infty/\ell_1$ sparse recovery in time $O(k^2 \log^3n)$. 2. We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein's inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of $\Omega(k^2 + k \log n)$ is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of $\Omega(k^2 \log n/ \log k)$ is known for incoherent matrices.Comment: ICALP 2020--presentation improved according to reviewers' comment

Federated Empirical Risk Minimization via Second-Order Method

Many convex optimization problems with important applications in machine learning are formulated as empirical risk minimization (ERM). There are several examples: linear and logistic regression, LASSO, kernel regression, quantile regression, $p$-norm regression, support vector machines (SVM), and mean-field variational inference. To improve data privacy, federated learning is proposed in machine learning as a framework for training deep learning models on the network edge without sharing data between participating nodes. In this work, we present an interior point method (IPM) to solve a general ERM problem under the federated learning setting. We show that the communication complexity of each iteration of our IPM is $\tilde{O}(d^{3/2})$, where $d$ is the dimension (i.e., number of features) of the dataset

Sparse Nonnegative Convolution Is Equivalent to Dense Nonnegative Convolution

Computing the convolution $A\star B$ of two length-$n$ vectors $A,B$ is an ubiquitous computational primitive. Applications range from string problems to Knapsack-type problems, and from 3SUM to All-Pairs Shortest Paths. These applications often come in the form of nonnegative convolution, where the entries of $A,B$ are nonnegative integers. The classical algorithm to compute $A\star B$ uses the Fast Fourier Transform and runs in time $O(n\log n)$. However, often $A$ and $B$ satisfy sparsity conditions, and hence one could hope for significant improvements. The ideal goal is an $O(k\log k)$-time algorithm, where $k$ is the number of non-zero elements in the output, i.e., the size of the support of $A\star B$. This problem is referred to as sparse nonnegative convolution, and has received considerable attention in the literature; the fastest algorithms to date run in time $O(k\log^2 n)$. The main result of this paper is the first $O(k\log k)$-time algorithm for sparse nonnegative convolution. Our algorithm is randomized and assumes that the length $n$ and the largest entry of $A$ and $B$ are subexponential in $k$. Surprisingly, we can phrase our algorithm as a reduction from the sparse case to the dense case of nonnegative convolution, showing that, under some mild assumptions, sparse nonnegative convolution is equivalent to dense nonnegative convolution for constant-error randomized algorithms. Specifically, if $D(n)$ is the time to convolve two nonnegative length-$n$ vectors with success probability $2/3$, and $S(k)$ is the time to convolve two nonnegative vectors with output size $k$ with success probability $2/3$, then $S(k)=O(D(k)+k(\log\log k)^2)$. Our approach uses a variety of new techniques in combination with some old machinery from linear sketching and structured linear algebra, as well as new insights on linear hashing, the most classical hash function

Traversing the FFT Computation Tree for Dimension-Independent Sparse Fourier Transforms

We consider the well-studied Sparse Fourier transform problem, where one aims to quickly recover an approximately Fourier $k$-sparse vector $\widehat{x} \in \mathbb{C}^{n^d}$ from observing its time domain representation $x$. In the exact $k$-sparse case the best known dimension-independent algorithm runs in near cubic time in $k$ and it is unclear whether a faster algorithm like in low dimensions is possible. Beyond that, all known approaches either suffer from an exponential dependence on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$ for $N=n^d$. Our work aims to address the above issues. First, we provide a translation/reduction of the exactly $k$-sparse FT problem to a concrete tree exploration task which asks to recover $k$ leaves in a full binary tree under certain exploration rules. Subsequently, we provide (a) an almost quadratic in $k$ time algorithm for this task, and (b) evidence that a strongly subquadratic time for Sparse FT via this approach is likely impossible. We achieve the latter by proving a conditional quadratic time lower bound on sparse polynomial multipoint evaluation (the classical non-equispaced sparse FT) which is a core routine in the aforementioned translation. Thus, our results combined can be viewed as an almost complete understanding of this approach, which is the only known approach that yields sublinear time dimension-independent Sparse FT algorithms. Subsequently, we provide a robustification of our algorithm, yielding a robust cubic time algorithm under bounded $\ell_2$ noise. This requires proving new structural properties of the recently introduced adaptive aliasing filters combined with a variety of new techniques and ideas

Sparse {Fourier Transform} by Traversing {Cooley-Tukey FFT} Computation Graphs

Computing the dominant Fourier coefficients of a vector is a common task in many fields, such as signal processing, learning theory, and computational complexity. In the Sparse Fast Fourier Transform (Sparse FFT) problem, one is given oracle access to a $d$-dimensional vector $x$ of size $N$, and is asked to compute the best $k$-term approximation of its Discrete Fourier Transform, quickly and using few samples of the input vector $x$. While the sample complexity of this problem is quite well understood, all previous approaches either suffer from an exponential dependence of runtime on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT algorithm of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$. In this work, we introduce a new high-dimensional Sparse FFT toolkit and use it to obtain new algorithms, both on the exact, as well as in the case of bounded $\ell_2$ noise. This toolkit includes i) a new strategy for exploring a pruned FFT computation tree that reduces the cost of filtering, ii) new structural properties of adaptive aliasing filters recently introduced by Kapralov, Velingker and Zandieh'SODA'19, and iii) a novel lazy estimation argument, suited to reducing the cost of estimation in FFT tree-traversal approaches. Our robust algorithm can be viewed as a highly optimized sparse, stable extension of the Cooley-Tukey FFT algorithm. Finally, we explain the barriers we have faced by proving a conditional quadratic lower bound on the running time of the well-studied non-equispaced Fourier transform problem. This resolves a natural and frequently asked question in computational Fourier transforms. Lastly, we provide a preliminary experimental evaluation comparing the runtime of our algorithm to FFTW and SFFT 2.0