Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real closed field and let $${\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$$ with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $\pi: \R^{\ell+k} \to \R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is bounded by $$(2^m\ell k d)^{O(mk)}.$$Comment: 27 pages, 1 figur

Sparse Regression via Range Counting

The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set S of n points in ?^d, a point y? ?^d, and an integer 2 ? k ? d, find an affine combination of at most k points of S that is nearest to y. We describe a O(n^{k-1} log^{d-k+2} n)-time randomized (1+?)-approximation algorithm for this problem with d and ? constant. This is the first algorithm for this problem running in time o(n^k). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP\u2718), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we provide a simple O_?(n^{d-1+?})-time deterministic exact algorithm, for any ? > 0. Finally, we show how to adapt the approximation algorithm for the sparse linear regression and sparse convex regression problems with the same running time, up to polylogarithmic factors

Fourier Growth of Structured ??-Polynomials and Applications

We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F?-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013]. - We show that any read-? degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? (k ? d)^{O(k)}. - We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F?-polynomials