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Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric…

Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is…

Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$…

We present two protocols for classical verification of quantum depth. Our protocols allow a purely classical verifier to distinguish devices with different quantum circuit depths even in the presence of classical computation. We show that a…

In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg. This class includes the famous identities by Ramanujan which provide a witness to the divisibility properties of $p(5n+4),$…

We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…

Current out-of-distribution (OOD) detection methods typically assume balanced in-distribution (ID) data, while most real-world data follow a long-tailed distribution. Previous approaches to long-tailed OOD detection often involve balancing…

Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus…

IFTT-PIN is a self-calibrating version of the PIN-entry method introduced in Roth et al. (2004) [1]. In [1], digits are split into two sets and assigned a color respectively. To communicate their digit, users press the button with the same…

We study the following computational problem: for which values of $k$, the majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula whose each gate computes a majority function of at most $k$ bits? The corresponding…

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which…

In recent years, finding new satisfiability algorithms for various circuit classes has been a very active line of research. Despite considerable progress, we are still far away from a definite answer on which circuit classes allow fast…

We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…

We examine the fundamental problem of constructing depth-optimum circuits for binary addition. More precisely, as in literature, we consider the following problem: Given auxiliary inputs $t_0, \dotsc, t_{m-1}$, so-called generate and…

Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be…

Fault injection attacks represent a type of active, physical attack against cryptographic circuits. Various countermeasures have been proposed to thwart such attacks, the design and implementation of which are, however, intricate,…

We propose an effective and robust algorithm for identifying partial differential equations (PDEs) with space-time varying coefficients from a single trajectory of noisy observations. Identifying unknown differential equations from noisy…

In differential privacy (DP), the generalized private testing problem was introduced by Liu and Talwar (STOC 2019). Given a dataset $X \in \mathcal{X}$ and a sequence of black-box $\varepsilon_t$-DP mechanisms $M_t:\mathcal{X}\to\{+1,-1\}$,…

In many science and engineering settings, system dynamics are characterized by governing PDEs, and a major challenge is to solve inverse problems (IPs) where unknown PDE parameters are inferred based on observational data gathered under…