arXiv subjects

Tasuku Soma

Publications and source records attributed to Tasuku Soma.

At least 19 recordsLinked to original sources

arXiv2026

Quiver Semistability and Structured Kalman Decompositions for Networked Linear Dynamical Systems

We introduce new notions of controllability and observability for networked linear time-invariant (LTI) systems based on $σ$-semistability of quiver representations. Utilizing King's criterion for $σ$-semistability, we define a network generalization of the Kalman decomposition for networked LTI systems, which systematically decomposes the local and interconnection dynamics while respecting the underlying network structure. Furthermore, we present efficient algorithms for deciding the proposed controllability and observability of a given networked LTI system and for finding the Kalman-type decomposition. We also show efficient algorithms for deciding the $σ$-semistability of representations of acyclic quivers with self-loops if the weight $σ$ has the same sign for all vertices with self-loops. Such quiver representations and weights arise from networked LTI systems.

math.OC↗

arXiv2026

Accelerating operator Sinkhorn iteration with overrelaxation

We propose accelerated versions of the operator Sinkhorn iteration for operator scaling using successive overrelaxation. We analyze the local convergence rates of these accelerated methods via linearization, which allows us to determine the asymptotically optimal relaxation parameter based on Young's SOR theorem. Using the Hilbert metric on positive definite cones, we also obtain a global convergence result for a geodesic version of overrelaxation in a specific range of relaxation parameters. These techniques generalize corresponding results obtained for matrix scaling by Thibault et al. (Algorithms, 14(5):143, 2021) and Lehmann et al. (Optim. Lett., 16(8):2209--2220, 2022). Numerical experiments demonstrate that the proposed methods outperform the original operator Sinkhorn iteration in certain applications.

math.OC↗

arXiv2026

Numerically stable variants of overrelaxation for operator Sinkhorn iteration

We consider accelerated versions of the operator Sinkhorn iteration (OSI) for solving scaling problems for completely positive maps. Based on the interpretation of OSI as alternating fixed point iteration, it has been recently proposed to achieve acceleration by means of nonlinear successive overrelaxation (SOR), e.g.~with respect to geodesics in Hilbert metric. The direct implementation of the proposed SOR algorithms, however, can be numerically unstable for ill-conditioned instances, limiting the achievable accuracy. Here we derive equivalent versions of OSI with SOR where, similar to the original OSI formulation, scalings are applied on the fly in order to take advantage of preconditioning effects. Numerical experiments confirm that this modification allows for numerically stable SOR-acceleration of OSI even in ill-conditioned cases.

math.OC↗

arXiv2025

$O(\log n)$-Approximation Algorithms for Bipartiteness Ratio

We propose an $O(\log n)$-approximation algorithm for the bipartiteness ratio of undirected graphs introduced by Trevisan (SIAM Journal on Computing, vol. 41, no. 6, 2012), where $n$ is the number of vertices. Our approach extends the cut-matching game framework for sparsest cut to the bipartiteness ratio, and requires only $\mathop{\mathrm{polylog}} n$ many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in almost linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartiteness ratio in terms of well-linkedness in an auxiliary skew-symmetric graph, which may be of independent interest. As an application, we devise an $\tilde{O}(mn)$-time algorithm for the minimum uncut problem: given a graph whose optimal cut leaves an $η$ fraction of edges uncut, we find a cut that leaves only an $O(\log n \log(1/η)) \cdot η$ fraction of edges uncut, where $m$ is the number of edges. Finally, we propose a directed analogue of the bipartiteness ratio, and we give a polynomial-time algorithm that achieves an $O(\log n)$ approximation for this measure via a directed Leighton--Rao-style embedding. We also propose an algorithm for the minimum directed uncut problem with a guarantee similar to that for the minimum uncut problem.

cs.DS↗

arXiv2025

Algorithmic aspects of semistability of quiver representations

We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and $σ$-semistability, finding the maximizers of King's criterion, and computing the Harder--Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King's criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the $σ$-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.

math.OC↗

arXiv2025

Near-Optimal Algorithms for Group Distributionally Robust Optimization and Beyond

Distributionally robust optimization (DRO) can improve the robustness and fairness of learning methods. In this paper, we devise stochastic algorithms for a class of DRO problems including group DRO, subpopulation fairness, and empirical conditional value at risk (CVaR) optimization. Our new algorithms achieve faster convergence rates than existing algorithms for multiple DRO settings. We also provide a new information-theoretic lower bound that implies our bounds are tight for group DRO. Empirically, too, our algorithms outperform known methods.

cs.LG↗

arXiv2024

Algebraic Algorithms for Fractional Linear Matroid Parity via Non-commutative Rank

Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix representations using the concept of non-commutative rank (nc-rank), which has recently attracted attention in the research of Edmonds' problem. We reveal that the nc-rank of the matrix representation of linear matroid parity corresponds to the optimal value of fractional linear matroid parity: a half-integral relaxation of linear matroid parity. Based on our representation, we present an algebraic algorithm for the fractional linear matroid parity problem by building a new technique to incorporate the search-to-decision reduction into the half-integral problem represented via the nc-rank. We further present a faster divide-and-conquer algorithm for finding a maximum fractional matroid matching and an algebraic algorithm for finding a dual optimal solution. They together lead to an algebraic algorithm for the weighted fractional linear matroid parity problem. Our algorithms are significantly simpler and faster than the existing algorithms.

math.OC↗

arXiv2023

Online Algorithms for Spectral Hypergraph Sparsification

We provide the first online algorithm for spectral hypergraph sparsification. In the online setting, hyperedges with positive weights are arriving in a stream, and upon the arrival of each hyperedge, we must irrevocably decide whether or not to include it in the sparsifier. Our algorithm produces an $(ε, δ)$-spectral sparsifier with multiplicative error $ε$ and additive error $δ$ that has $O(ε^{-2} n \log n \log r \log(1 + εW/δn))$ hyperedges with high probability, where $ε, δ\in (0,1)$, $n$ is the number of nodes, and $W$ is the sum of edge weights. The space complexity of our algorithm is $O(n^2)$, while previous algorithms require the space complexity of $Ω(m)$, where $m$ is the number of hyperedges. This provides an exponential improvement in the space complexity since $m$ can be exponential in $n$.

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arXiv2023

Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

We address the computation of the degrees of minors of a noncommutative symbolic matrix of form \[ A[c] := \sum_{k=1}^m A_k t^{c_k} x_k, \] where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes.

math.CO↗

arXiv2022

Shrunk subspaces via operator Sinkhorn iteration

A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope.

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arXiv2021

Online Risk-Averse Submodular Maximization

We present a polynomial-time online algorithm for maximizing the conditional value at risk (CVaR) of a monotone stochastic submodular function. Given $T$ i.i.d. samples from an underlying distribution arriving online, our algorithm produces a sequence of solutions that converges to a ($1-1/e$)-approximate solution with a convergence rate of $O(T^{-1/4})$ for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require $Ω(T)$ space, our online algorithm only requires $O(\sqrt{T})$ space. We extend our online algorithm to portfolio optimization for monotone submodular set functions under a matroid constraint. Experiments conducted on real-world datasets demonstrate that our algorithm can rapidly achieve CVaRs that are comparable to those obtained by existing offline algorithms.

cs.DS↗

arXiv2021

Information geometry of operator scaling

Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. Other types of alternating e-projections algorithms are also provided by using different information geometric structures on the positive definite cone.

math.OC↗

arXiv2020

Statistical Learning with Conditional Value at Risk

We propose a risk-averse statistical learning framework wherein the performance of a learning algorithm is evaluated by the conditional value-at-risk (CVaR) of losses rather than the expected loss. We devise algorithms based on stochastic gradient descent for this framework. While existing studies of CVaR optimization require direct access to the underlying distribution, our algorithms make a weaker assumption that only i.i.d.\ samples are given. For convex and Lipschitz loss functions, we show that our algorithm has $O(1/\sqrt{n})$-convergence to the optimal CVaR, where $n$ is the number of samples. For nonconvex and smooth loss functions, we show a generalization bound on CVaR. By conducting numerical experiments on various machine learning tasks, we demonstrate that our algorithms effectively minimize CVaR compared with other baseline algorithms.

cs.LG↗

arXiv2018

Fast greedy algorithms for dictionary selection with generalized sparsity constraints

In dictionary selection, several atoms are selected from finite candidates that successfully approximate given data points in the sparse representation. We propose a novel efficient greedy algorithm for dictionary selection. Not only does our algorithm work much faster than the known methods, but it can also handle more complex sparsity constraints, such as average sparsity. Using numerical experiments, we show that our algorithm outperforms the known methods for dictionary selection, achieving competitive performances with dictionary learning algorithms in a smaller running time.

cs.LG↗

arXiv2018

No-regret algorithms for online $k$-submodular maximization

We present a polynomial time algorithm for online maximization of $k$-submodular maximization. For online (nonmonotone) $k$-submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone $k$-submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as $k$ tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.

cs.DS↗

arXiv2018

Spectral Sparsification of Hypergraphs

For an undirected/directed hypergraph $G=(V,E)$, its Laplacian $L_G\colon\mathbb{R}^V\to \mathbb{R}^V$ is defined such that its ``quadratic form'' $\boldsymbol{x}^\top L_G(\boldsymbol{x})$ captures the cut information of $G$. In particular, $\boldsymbol{1}_S^\top L_G(\boldsymbol{1}_S)$ coincides with the cut size of $S \subseteq V$, where $\boldsymbol{1}_S \in \mathbb{R}^V$ is the characteristic vector of $S$. A weighted subgraph $H$ of a hypergraph $G$ on a vertex set $V$ is said to be an $ε$-spectral sparsifier of $G$ if $(1-ε)\boldsymbol{x}^\top L_H(\boldsymbol{x}) \leq \boldsymbol{x}^\top L_G(\boldsymbol{x}) \leq (1+ε)\boldsymbol{x}^\top L_H(\boldsymbol{x})$ holds for every $\boldsymbol{x} \in \mathbb{R}^V$. In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph $G$ on $n$ vertices, constructs an $ε$-spectral sparsifier of $G$ with $O(n^3\log n/ε^2)$ hyperedges/hyperarcs. The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of $L_G$, computing the effective resistance between a pair of vertices in $G$, semi-supervised learning based on $L_G$, and cut problems on $G$. In addition, our sparsification result implies that any submodular function $f\colon 2^V \to \mathbb{R}_+$ with $f(\emptyset)=f(V)=0$ can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions $f\colon 2^V \to [0,1]$ with $f(\emptyset)=f(V)=0$, with $O(n^4\log (n/ε) /ε^4)$ samples.

cs.DS↗

arXiv2018

Maximally Invariant Data Perturbation as Explanation

While several feature scoring methods are proposed to explain the output of complex machine learning models, most of them lack formal mathematical definitions. In this study, we propose a novel definition of the feature score using the maximally invariant data perturbation, which is inspired from the idea of adversarial example. In adversarial example, one seeks the smallest data perturbation that changes the model's output. In our proposed approach, we consider the opposite: we seek the maximally invariant data perturbation that does not change the model's output. In this way, we can identify important input features as the ones with small allowable data perturbations. To find the maximally invariant data perturbation, we formulate the problem as linear programming. The experiment on the image classification with VGG16 shows that the proposed method could identify relevant parts of the images effectively.

stat.ML↗

arXiv2018

Polynomial-Time Algorithms for Submodular Laplacian Systems

Let $G=(V,E)$ be an undirected graph, $L_G\in \mathbb{R}^{V \times V}$ be the associated Laplacian matrix, and $b \in \mathbb{R}^V$ be a vector. Solving the Laplacian system $L_G x = b$ has numerous applications in theoretical computer science, machine learning, and network analysis. Recently, the notion of the Laplacian operator $L_F:\mathbb{R}^V \to 2^{\mathbb{R}^V}$ for a submodular transformation $F:2^V \to \mathbb{R}_+^E$ was introduced, which can handle undirected graphs, directed graphs, hypergraphs, and joint distributions in a unified manner. In this study, we show that the submodular Laplacian system $L_F( x) \ni b$ can be solved in polynomial time. Furthermore, we also prove that even when the submodular Laplacian system has no solution, we can solve its regression form in polynomial time. Finally, we discuss potential applications of submodular Laplacian systems in machine learning and network analysis.

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