arXiv subjects

Makoto Yamashita

Publications and source records attributed to Makoto Yamashita.

At least 19 recordsLinked to original sources

arXiv2026

A One-Factorization Predictor-Corrector Long-Step Arc-Search Method and a Curvature-Amplified Variant for Semidefinite Programming with a Homogeneous Self-Dual Embedding

In a predictor-corrector arc-search method, a point on the predictor arc is first selected and a corrector is then computed at that point. Since the Karush-Kuhn-Tucker (KKT) matrix changes with the selected point, this correction may require a second factorization in each iteration. We propose a one-factorization arc-search method (OFAS) for semidefinite programming (SDP) that avoids this second factorization, retains the corrector after the arc step, and uses only one KKT factorization per iteration. The key is a three-term decomposition of the complementarity mismatch at the selected point on the arc, which allows the correction to be assembled from linear-system solutions using the factorization at the current iterate. We also propose a curvature-amplified variant (CA-OFAS) that amplifies the second-order arc term. The proposed methods are formulated with a homogeneous self-dual embedding. For a wide long-step neighborhood, we prove that both methods reduce the homogeneous complementarity and residuals below $\varepsilon$ in $O(n \log(1/\varepsilon))$ iterations for an $n$-by-$n$ SDP matrix variable. Numerical experiments show that the one-factorization structure and the correction after the arc step are associated with reductions in computation time. Compared with the Mehrotra-type method, CA-OFAS reduces the geometric-mean computation time by 17% on SDPLIB and 13% on neural network verification SDP problems.

math.OC↗

arXiv2026

Order-2 Tightness of Block-Sparse SOS Relaxations for One-Layer ReLU Network Verification with a Matching Input-Sharing Graph

Azuma, Kim, and Yamashita formulated the verification problem for one-layer ReLU networks as a quadratically constrained quadratic program and established tight semidefinite relaxations for the edgeless case and for one-unit settings. In this work, we represent the sharing pattern of undecided ReLUs over a box input set through an input-sharing graph and focus on the case where this graph is a matching. We then derive an explicit, checkable sufficient condition for the tightness of the order-$2$ block-sparse SOS relaxation associated with the connected-component decomposition of this graph. Under the matching assumption, the global problem decomposes into isolated-vertex blocks and single-edge blocks. The key difficulty, which is absent from the edgeless case, is establishing tightness for a two-unit edge block. For regular rank-one edges, we show that the convex hull of each two-unit local set can be described exactly by two reduced one-unit hulls coupled through a common shared scalar. Combining the one-unit tightness result of Azuma et al. with Farkas' lemma and affine elimination, we obtain a local order-$2$ certificate for each edge block. Isolated-vertex blocks reduce to one-unit problems over box input sets and are therefore handled at the same order. We prove that, when the input-sharing graph is a matching and every edge satisfies the regular rank-one condition, the order-$2$ block-sparse SOS relaxation is tight. This extends the tight sparse relaxation result for the edgeless case to the first sparse setting with a nontrivial two-unit interaction.

math.OC↗

arXiv2026

Tight Conic Relaxations for Rank-one Doubly Nonnegative Matrix Completion

We study tight conic relaxations for a quadratically constrained quadratic programming (QCQP) formulation of rank-one doubly nonnegative (DNN) matrix completion. Motivated by sparse QCQPs whose lifted matrix variables include elements not directly specified by the objective or constraints, we interpret tightness as a rank-one completion property for the unspecified elements. For sparsity patterns whose blocks consist of cycles and edges, we prove that the dual formulations associated with the DNN and completely positive (CP) relaxations are equivalent. For cycle-type sparsity patterns, we derive explicit sufficient conditions under which the semidefinite programming (SDP) and DNN relaxations are tight. These sufficient conditions are stated explicitly in terms of local ratio bounds and cumulative-difference conditions on a rank-one optimal solution. We also show that adding suitable edges to the sparsity pattern relaxes the ratio conditions required for tightness. The results provide tractable certificates for when conic relaxations recover a rank-one optimal solution of the underlying QCQP.

math.OC↗

arXiv2026

A Benchmark of 25 Nonlinear Functions with Domain-Induced Discontinuity-Like for Global Optimization

A benchmark of 25 nonlinear optimization problems exhibiting domain-induced discontinuity-like behavior is proposed to support the performance evaluation of global optimization algorithms under feasibility-scarce and structurally disconnected landscapes. Referred to as the CPC Benchmark (Challenging Problems for Computation), the test suite consists of functions that are continuous on their natural domains, while infeasible regions and undefined evaluations are implicitly embedded in the objective, creating substantial challenges for global minimization. Six representative algorithms from diverse methodological paradigms are assessed to investigate the structural difficulty and discriminative capability of the proposed benchmark. Numerical results show that many functions possess extremely small feasible regions and strong precision sensitivity near feasibility boundaries, complicating initialization, feasibility discovery, and reliable objective evaluation. The findings demonstrate that the CPC benchmark provides a rigorous and discriminative testbed for advancing research on global optimization under hidden feasibility constraints.

math.OC↗

arXiv2026

Chern character for torsion-free ample groupoids

For an ample groupoid with torsion-free stabilizers, we construct a Chern character map going from the domain of the Baum-Connes assembly map of G to the Crainic-Moerdijk homology groups of G with rational coefficients. Assuming the (rational) Baum-Connes conjecture, this implies that the operator K-groups of the groupoid C*-algebra are rationally isomorphic to the periodicized groupoid homology groups, confirming a variation of Matui's HK conjecture. Our construction hinges on the recent $\infty$-categorical viewpoint on bivariant K-theory. We also present applications to the homology of hyperbolic dynamical systems, the homology of topological full groups, the homotopy type of the algebraic K-theory spectrum of ample groupoids, and the Elliott invariant of classifiable C*-algebras.

math.KT↗

arXiv2026

Categorical dualtiy for Yetter-Drinfeld C*-algebras. Beyond the braided-commutative case

We develop a tensor categorical duality in the sprit of the Tannaka-Krein duality for the C*-algebras admitting the Yetter-Drinfeld module structure over a compact quantum group. Under this duality, given a reduced compact quantum group G, the Yetter-Drinfeld G-C*-algebras correspond to the bimodule categories over the representation category Rep(G), satisfying a certain centrality condition.

math.OA↗

arXiv2026

An inexact infeasible arc-search interior-point method for linear optimization problems

We propose an inexact infeasible arc-search interior-point method for solving linear optimization problems. The method combines an arc-search strategy with inexact solutions to Newton systems and admits a polynomial iteration complexity bound. In existing inexact infeasible interior-point methods, both the linearization error of the central path and the inexactness of the Newton system accumulate along the search direction, which forces the algorithm to take very small steps. The proposed method mitigates this effect by using an arc-search strategy: the curved search path provides a more accurate approximation of the central path, so the step size can remain larger even when the Newton system is solved inexactly. As a result, the proposed method achieves a provably tighter worst-case iteration bound than existing inexact infeasible line-search methods. Numerical experiments on NETLIB benchmark problems demonstrate that the proposed method reduces both the number of iterations and the computation time.

math.OC↗

arXiv2025

Waveguide-array-based multiplexed photonic interface for atom array

The growing demand for high-capacity quantum communication and large-scale quantum computing underscores the importance of networking quantum processing units via multiplexed photonic channels. A neutral atom array with multiplexed atom-photon entanglement is a promising platform for its realization. Here, we demonstrate a key multiplexed photonic interface guiding the photons from an atom array to a single-mode waveguide array fabricated on a glass-based photonic integrated circuit. Remarkable 10 channels out of the 32-channel waveguide array with 25 $μ$m pitch couple to photons from 10 sites of the atom array with Rydberg gate-enabled separation. Based on the observed correlation between the atomic states and the polarization of the photon with a visibility of 0.87, we anticipate its applicability to a large-scale multiplexed atom-photon entanglement generation for networking quantum processing units.

quant-ph↗

arXiv2025

The braided monoidal structure of tube algebra representations

We consider the tube algebra of a spherical semisimple multitensor category $\mathcal{X}$, and construct a braided monoidal structure with twist for its representations. We further show that this category is braided tensor equivalent with the Drinfeld center of the ind-category of $\mathcal{X}$, extending the well-known linear equivalence.

math.QA↗

arXiv2025

On the structure of DHR bimodules of abstract spin chains

Abstract spin chains axiomatize the structure of local observables on the 1D lattice which are invariant under a global symmetry, and arise at the physical boundary of 2+1D topologically ordered spin systems. In this paper, we study tensor categorical properties of DHR bimodules over abstract spin chains. Assuming that the charge transporters generate the algebra of observables, we prove that the associated category has a structure of modular tensor category with respect to the natural braiding. Under an additional assumption of algebraic Haag duality, this category becomes the Drinfeld center of the half-line fusion category.

math.QA↗

arXiv2025

Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for groupoids which are principal with orbits homeomorphic to a Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

math.KT↗

arXiv2025

Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces

We study homological invariants of étale groupoids arising from Smale spaces, continuing on our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology groups defined by Putnam are isomorphic to the Crainic-Moerdijk groupoid homology with integer coefficients. We also show that the K-groups of C*-algebras of stable and unstable equivalence relations have finite rank. For unstably disconnected Smale spaces, we provide a cohomological spectral sequence whose second page is the (stable) homology groups, and converges to the K-groups of the unstable C*-algebra.

math.KT↗

arXiv2025

Poisson-Lie Group structures on semidirect products

We look at the Poisson structure on the total space of the dual bundle to the Lie algebroid arising from a matched pair of Lie groups. This dual bundle, with the natural semidirect product group structure, becomes a Poisson-Lie group as suggested by a recent work of Stachura. Moreover, when we start from matched pairs given by the Iwasawa decomposition of simple Lie groups, the associated Lie bialgebra is coboundary.

math.QA↗

arXiv2025

Quantum symmetry of $3$-transitive graphs

We study the quantum automorphism group of $3$-transitive graphs in this article. Those are highly symmetric graphs that were classified by Cameron and Macpherson in 1985, and we compute the quantum automorphism group of all such graphs, excluding the orthogonal graphs $\mathrm{O}^-(6,q)$ for $q>3$. We show that there is no quantum symmetry for the McLaughlin graph and the orthogonal graphs $\mathrm{O}^-(6,q)$ with $q = 2, 3$, while that the quantum automorphism group of the affine polar graphs $\mathrm{VO}^{+}(2k,2)$ and $\mathrm{VO}^{-}(2k,2)$ are monoidally equivalent to $\mathrm{PO}(n)$ and $\mathrm{PSp}(n)$, respectively. We use planar algebras to obtain our results, where the $3$-transitivity of the graphs gives bounds on the dimensions of the $2$-- and $3$-box spaces of the associated planar algebras.

math.QA↗

arXiv2025

Injectivity for algebras and categories with quantum symmetry

We establish the existence of injective envelopes for unital Yetter-Drinfeld C*-algebras, and a related class of bimodule categories over rigid C*-tensor categories. This implies monoidal invariance for boundary actions of Drinfeld doubles of compact quantum groups.

math.OA↗

arXiv2025

Tight Semidefinite Relaxations for Verifying Robustness of Neural Networks

For verifying the safety of neural networks (NNs), Fazlyab et al. (2019) introduced a semidefinite programming (SDP) approach called DeepSDP. This formulation can be viewed as the dual of the SDP relaxation for a problem formulated as a quadratically constrained quadratic program (QCQP). While SDP relaxations of QCQPs generally provide approximate solutions with some gaps, this work focuses on tight SDP relaxations that provide exact solutions to the QCQP for single-layer NNs. Specifically, we analyze tightness conditions in three cases: (i) NNs with a single neuron, (ii) single-layer NNs with an ellipsoidal input set, and (iii) single-layer NNs with a rectangular input set. For NNs with a single neuron, we propose a condition that ensures the SDP admits a rank-1 solution to DeepSDP by transforming the QCQP into an equivalent two-stage problem leads to a solution collinear with a predetermined vector. For single-layer NNs with an ellipsoidal input set, the collinearity of solutions is proved via the Karush-Kuhn-Tucker condition in the two-stage problem. In case of single-layer NNs with a rectangular input set, we demonstrate that the tightness of DeepSDP can be reduced to the single-neuron NNs, case (i), if the weight matrix is a diagonal matrix.

math.OC↗

arXiv2025

Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez-Etingof cyclotomic Knizhnik-Zamolodchikov (KZ) equations and the other on the Letzter-Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno-Drinfeld type theorem on type B braid group representations defined by the monodromy of KZ-equations and by the Balagović-Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

math.QA↗

arXiv2024

Crystallization of C*-algebras

Given a C$^*$-algebra $A$ with an almost periodic time evolution $σ$, we define a new C$^*$-algebra $A_c$, which we call the crystal of $(A,σ)$, that represents the zero temperature limit of $(A, σ)$. We prove that there is a one-to-one correspondence between the ground states of $(A,σ)$ and the states on $A_c$, justifying the name. In order to investigate further the relation between low temperature equilibrium states on $A$ and traces on $A_c$, we define a Fock module $\mathcal F$ over the crystal and construct a vacuum representation of $A$ on $\mathcal F$. This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures $β$ the $σ$-KMS$_β$-states on $A$ are induced from traces on $A_c$ by means of the Fock module. In the second part, we compare the K-theoretic structures of $A$ and $A_c$. Previous work by various authors suggests that they have (rationally) isomorphic K-groups. We analyze this phenomenon in detail, confirming it under favorable conditions, but showing that, in general, there is apparently no easy way to relate these groups. As examples, we discuss in particular Exel's results on semi-saturated circle actions, and recent results of Miller on the K-theory of inverse semigroup C$^*$-algebras. In relation to the latter, we introduce the notion of a scale $N$ on an inverse semigroup $I$ and define a new inverse semigroup $I_c$, which we call the crystal of $(I,N)$.

math.OA↗