arXiv subjects

Jianting Yang

Publications and source records attributed to Jianting Yang.

8 recordsLinked to original sources

arXiv2026

Noncommutative Rational Sums of Squares in Free Algebras

This paper introduces rational sums of squares for symmetric noncommutative polynomials in free algebras. Under the assumption that the polynomial admits a strictly positive scalar self-adjoint evaluation, our main result gives an operator-theoretic characterization of this class: a polynomial is a rational sum of squares if and only if there exists a degree bound such that, for every self-adjoint operator evaluation satisfying a natural nondegeneracy condition, the compression of the evaluated polynomial to the associated finite-dimensional cyclic subspace is not negative definite. We also prove that sums of Hermitian squares form a proper subset of rational sums of squares when the free algebra has at least two generators, whereas the two classes coincide for homogeneous polynomials. Finally, we show that the set of rational sums of squares is nonconvex when there are at least three generators, and that its complement is also nonconvex.

math.RA↗

arXiv2026

A Homogeneous Nullstellensatz for Joint Invariant Subspaces

Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the unital $\mathbb{K}$-algebra generated by finitely many noncommutative polynomials if and only if, for matrices of every size, every joint invariant subspace of the evaluations of the generators is also invariant under the evaluation of $g$. In this paper, we establish a homogeneous Nullstellensatz for joint invariant subspaces by proving that this equivalence holds whenever the generators are homogeneous. In contrast, we demonstrate that the statement fails in the general case, thereby settling the conjecture completely.

math.RA↗

arXiv2025

A Counterexample to the Optimality Conjecture in Convex Quantum Channel Optimization

This paper presents a counterexample to the optimality conjecture in convex quantum channel optimization proposed by Coutts et al. The conjecture posits that for nuclear norm minimization problems in quantum channel optimization, the dual certificate of an optimal solution can be uniquely determined via the spectral calculus of the Choi matrix. By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture.

quant-ph↗

arXiv2024

Verifying Properties of Binary Neural Networks Using Sparse Polynomial Optimization

This paper explores methods for verifying the properties of Binary Neural Networks (BNNs), focusing on robustness against adversarial attacks. Despite their lower computational and memory needs, BNNs, like their full-precision counterparts, are also sensitive to input perturbations. Established methods for solving this problem are predominantly based on Satisfiability Modulo Theories and Mixed-Integer Linear Programming techniques, which are characterized by NP complexity and often face scalability issues. We introduce an alternative approach using Semidefinite Programming relaxations derived from sparse Polynomial Optimization. Our approach, compatible with continuous input space, not only mitigates numerical issues associated with floating-point calculations but also enhances verification scalability through the strategic use of tighter first-order semidefinite relaxations. We demonstrate the effectiveness of our method in verifying robustness against both $\|.\|_\infty$ and $\|.\|_2$-based adversarial attacks.

cs.LG↗

arXiv2023

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic theory of representation theory of finite groups indicates that a function $f$ on a finite abelian group $G$ can be written as a linear combination of characters of irreducible representations of $G$ by $ f(x)=\sum_{χ\in \widehat{G}} \widehat{f} (χ)χ(x)$, where $\widehat{G}$ is the dual group of $G$ consisting of all characters of $G$ and $ \widehat{f} (χ)$ is the Fourier coefficient of $f$ at $χ\in \widehat{G}$. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of $f$ on a finite abelian group $G$ with complexity \if $\operatorname{O}\left(|G| \log(|G|)+\log(k_{\min})\operatorname{SDP}(2k_{\min})\right)$,\fi that is quasi-linear in the order of $G$ and polynomial in the FSOS sparsity \if $k_{\min}$\fi of $f$. Moreover, for a nonnegatvie function $f$ on a finite abelian group $G$ and a set $S \subset \widehat{G}$, we give a lower bound of the constant $M$ such that $f+M$ admits an FSOS supported on} $S$. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to $10^7$. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem \if on $\mathbb{T}^n$\fi by sparse FSOS.

math.OC↗

arXiv2023

The non-Archimedean Nirgendsnegativsemidefinitheitsstellensatz is not true

Klep and Schweighofer asked whether the Nirgendsnegativsemide-finitheitsstellensatz holds for a symmetric noncommutative polynomial whose evaluations at bounded self-adjoint operators on any nontrivial Hilbert space are not negative semidefinite. We provide an example to show the open problem has a negative answer.

math.OA↗

arXiv2023

A Characterization of Perfect Strategies for Mirror Games

We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gröbner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.

math.OA↗

arXiv2022

Sparse sum of Hermitian squares in group algebras of finite groups

Nonnegative elements in group algebras play a central role in harmonic analysis, operator algebras, and computational optimization. This paper investigates sparse sum-of-Hermitian-squares (SOHS) representations of nonnegative elements in the group algebras of finite groups. We prove that the convex relaxation of the sparse SOHS problem admits a closed-form solution, namely the square root of the given element. Based on this result, we propose a thresholding hierarchy for approximating sparse SOHS representations. We analyze the error of this hierarchy with respect to two natural residuals and establish exponential decay rates. Remarkably, one error bound is independent of the group order, and the other is also group-size independent when the group is cyclic or dihedral. These results extend existing work on Fourier sums of squares for abelian groups to a broader class of finite groups and provide new algorithmic tools for sparse noncommutative positivity certificates.

math.OC↗