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The effective utilization of structural information in data while ensuring statistical validity poses a significant challenge in false discovery rate (FDR) analyses. Conformal inference provides rigorous theory for grounding complex machine…
Solving Partial Differential Equations (PDEs) is the core of many fields of science and engineering. While classical approaches are often prohibitively slow, machine learning models often fail to incorporate complete system information.…
An exciting recent development is the uptake of deep neural networks in many scientific fields, where the main objective is outcome prediction with the black-box nature. Significance testing is promising to address the black-box issue and…
A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ is nonzero. In this paper, we give…
We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to…
We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any…
Solving partial differential equations (PDEs) has been indispensable in scientific and engineering applications. Recently, deep learning methods have been widely used to solve high-dimensional problems, one of which is the physics-informed…
Recently, a polynomial time classical algorithm has been found for learning the shallow representation of a unitary $U$ acting on $n$-qubits, by learning local inversions and then sewing them back together with ancilla qubits and SWAP…
We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising…
In this work, we prove the strongest known lower bounds for QAC$^0$, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC$^0$ circuits cannot compute PARITY, and require $\Omega(\exp(\sqrt{n}))$ gates…
Deep Learning (DL) systems are increasingly deployed in safety-critical applications, yet they remain vulnerable to robustness issues that can lead to significant failures. While numerous Test Input Generators (TIGs) have been developed to…
One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing…
We study the complexity of securely evaluating arithmetic circuits over finite rings. This question is motivated by natural secure computation tasks. Focusing mainly on the case of two-party protocols with security against malicious…
The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closed-form solutions only exist for very simple differential equations, proof certificates have been proposed for more…
We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clifford+T gate set. First we construct a special subset of T-depth 1 unitaries, such that it is possible to express the T-depth-optimal decomposition of…
Let $S_d(n)$ denote the minimum number of wires of a depth-$d$ (unbounded fan-in) circuit encoding an error-correcting code $C:\{0, 1\}^n \to \{0, 1\}^{32n}$ with distance at least $4n$. G\'{a}l, Hansen, Kouck\'{y}, Pudl\'{a}k, and Viola…
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the…
Given a random $n \times n$ symmetric matrix $\boldsymbol W$ drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form $\boldsymbol x^\top \boldsymbol…
Checking whether two quantum circuits are approximately equivalent is a common task in quantum computing. We consider a closely related identity check problem: given a quantum circuit $U$, one has to estimate the diamond-norm distance…
Ising machines can solve combinatorial optimization problems by representing them as energy minimization problems. A common implementation is the probabilistic Ising machine (PIM), which uses probabilistic (p-) bits to represent coupled…