Related papers: Degree-$d$ Chow Parameters Robustly Determine Degr…

In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must…

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…

Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…

We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for…

Multi-index models - functions which only depend on the covariates through a non-linear transformation of their projection on a subspace - are a useful benchmark for investigating feature learning with neural nets. This paper examines the…

We study the probabilistic degree over reals of the OR function on $n$ variables. For an error parameter $\epsilon$ in (0,1/3), the $\epsilon$-error probabilistic degree of any Boolean function $f$ over reals is the smallest non-negative…

As datasets grow richer, an important challenge is to leverage the full features in the data to maximize the number of useful discoveries while controlling for false positives. We address this problem in the context of multiple hypotheses…

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…

Many applications that use empirically estimated functions face a curse of dimensionality, because the integrals over most function classes must be approximated by sampling. This paper introduces a novel regression-algorithm that learns…

The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : \{0,1\}^n \to \{0,1\}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps…

The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a…

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is a codeword of the Reed-Muller code of dimension $n$ and order $d$ over the finite field $\mathbb{F}_q$ for prime $q$…

Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood…

One of the arguments to explain the success of deep learning is the powerful approximation capacity of deep neural networks. Such capacity is generally accompanied by the explosive growth of the number of parameters, which, in turn, leads…

Embedding-based representations in Euclidean space $\mathbb{R}^d$ are a cornerstone of modern machine learning, where a major goal is to use the \emph{smallest dimension} that faithfully captures data relations. In this work, we prove sharp…

Classification of multi-dimensional time series from real-world systems require fine-grained learning of complex features such as cross-dimensional dependencies and intra-class variations-all under the practical challenge of low training…

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the…

In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives…

Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular…

We study the $\textit{average-case deterministic query complexity}$ of boolean functions under a $\textit{uniform input distribution}$, denoted by $\mathrm{D}_\mathrm{ave}(f)$, the minimum average depth of zero-error decision trees that…