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We give an algorithm that learns arbitrary Boolean functions of $k$ arbitrary halfspaces over $\mathbb{R}^n$, in the challenging distribution-free Probably Approximately Correct (PAC) learning model, running in time $2^{\sqrt{n} \cdot (\log…
Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over…
This article studies the achievable guarantees on the error rates of certain learning algorithms, with particular focus on refining logarithmic factors. Many of the results are based on a general technique for obtaining bounds on the error…
In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is…
Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis…
In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization.…
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial…
Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x…
Abstract notions of convexity over the vertices of a graph, and corresponding notions of halfspaces, have recently gained attention from the machine learning community. In this work we study monophonic halfspaces, a notion of graph…
Optimizing and certifying the positivity of polynomials are fundamental primitives across mathematics and engineering applications, from dynamical systems to operations research. However, solving these problems in practice requires large…
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty…
The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on…
Several recent works [DHLNSY25, CPPS25a, CPPS25b] have studied a model of property testing of Boolean functions under a \emph{relative-error} criterion. In this model, the distance from a target function $f: \{0,1\}^n \to \{0,1\}$ that is…
We prove convergence rates of linear sampling recovery of functions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algorithm is a function-valued…
This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional…
We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on…
Replacing non-polynomial functions (e.g., non-linear activation functions such as ReLU) in a neural network with their polynomial approximations is a standard practice in privacy-preserving machine learning. The resulting neural network,…
In large-data applications, such as the inference process of diffusion models, it is desirable to design sampling algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of sampling, which is the…
We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with…