arXiv subjects
Publications and source records attributed to Donggyu Kim.
Let $p_k(n)$ be the number of $k$-polymatroids on $[n]$. We show that for every fixed $k \geq 1$, we have \[ \left\lfloor \frac{k}{2} \right\rfloor \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)) \le \log_2 p_k(n) \le k \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)). \] We also show that for $k \geq 2$, almost all $k$-polymatroids are (i) connected, (ii) proper, and (iii) not linearly representable over any field.
While pretrained Vision-Language-Action (VLA) models offer broad generalization capabilities in robotic manipulation tasks, adapting them to real-world environments or handling task shifts often requires substantial additional data and retraining. To address this, we propose Symmetric VLA Distillation (SymVD), a distillation framework that transfers knowledge from a large VLA teacher to a compact student policy by explicitly exploiting geometric symmetries in manipulation tasks, such as rotational and reflectional invariance. SymVD employs an equivariant actor-critic architecture and trains the student using a symmetry-aware objective that aligns with teacher actions under group-invariant properties. We demonstrate that by enforcing the policy to respect equivariance, SymVD reduces redundant exploration across configurations related by group transformations and improves sample efficiency during distillation. To further stabilize and improve distillation, SymVD introduces an adaptive weighting scheme that dynamically balances the distillation objective and reinforcement learning updates based on training progress, enabling robust transfer even when the teacher signal is imperfect or misaligned. Experimental results on robotic manipulation tasks demonstrate that SymVD consistently improves over standard distillation and also outperforms SAC in terms of sample efficiency and generalization to previously unseen symmetric transformations of the environment.
Inhomogeneous broadening and spatial gradients in control fields inevitably produce large variations in characteristic sensitivity across spin ensembles. We derive an analytic expression for the sensitivity of an inhomogeneous ensemble and introduce a sensitivity threshold that identifies the optimal subset of spins. For both pulsed and continuous-wave magnetometry, the optimal ensembles deliver up to an eightfold improvement over conventional schemes relying on nominally uniform regions of the ensemble. We demonstrate phase-only digital holography to implement the optimal ensembles and show that the measured illumination non-uniformity limits the sensitivity by 14%. This approach enables optimal quantum sensing in optically scattering environments.
Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.
One of the most important classes of even $Δ$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong $Δ$-matroids and even $Δ$-matroids due to Geelen and Murota, we characterize the class of strong $Δ$-matroids that correspond to orientable ribbon-graphic $Δ$-matroids. These are precisely the $Δ$-matroids associated with what we call pseudo-orientable ribbon graphs. Moreover, we present a geometric construction that transforms a pseudo-orientable ribbon graph into an orientable ribbon graph, thereby realizing this correspondence. As consequences, we obtain the Matrix--Quasi-tree Theorem, the Hurwitz stability of quasi-tree generating polynomials, and a log-concavity result for the sequence counting quasi-trees of size $2i-1$ or $2i$ for pseudo-orientable ribbon graphs. To establish the log-concavity, we generalize Stanley's log-concavity theorem for regular matroids to regular $Δ$-matroids. Finally, we exhibit an infinite family of non-pseudo-orientable ribbon graphs that fail to satisfy the Matrix--Quasi-tree theorem and Hurwitz stability.
We give a new proof of Maurer's homotopy theorem for matroids using polyhedral methods, in contrast to Maurer's original combinatorial proof. The same polyhedral approach yields a homotopy theorem for delta-matroids, from which the corresponding results for matroids, even delta-matroids, and antisymmetric matroids follow. We further prove an analogous theorem for integral polymatroids.
Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.
In this paper, we propose a novel high-dimensional time-varying coefficient estimator for noisy high-frequency observations with a factor structure. In high-frequency finance, we often observe that noises dominate the signal of underlying true processes and that covariates exhibit a factor structure due to their strong dependence. Thus, we cannot apply usual regression procedures to analyze high-frequency observations. To handle the noises, we first employ a smoothing method for the observed dependent and covariate processes. Then, to handle the strong dependence of the covariate processes, we apply Principal Component Analysis (PCA) and transform the highly correlated covariate structure into a weakly correlated structure. However, the variables from PCA still contain non-negligible noises. To manage these non-negligible noises and the high dimensionality, we propose a nonconvex penalized regression method for each local coefficient. This method produces consistent but biased local coefficient estimators. To estimate the integrated coefficients, we propose a debiasing scheme and obtain a debiased integrated coefficient estimator using debiased local coefficient estimators. Then, to further account for the sparsity structure of the coefficients, we apply a thresholding scheme to the debiased integrated coefficient estimator. We call this scheme the Factor Adjusted Thresholded dEbiased Nonconvex LASSO (FATEN-LASSO) estimator. Furthermore, this paper establishes the concentration properties of the FATEN-LASSO estimator and discusses a nonconvex optimization algorithm.
Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann--Plücker functions. The equivalence is motivated by Cayley's identities expressing principal and almost-principal minors of a skew-symmetric matrix in terms of its Pfaffians. As a corollary of the new cryptomorphism, we deduce that each component of the orthogonal Grassmannian is parameterized by certain part of the Plücker coordinates.
In this paper, we develop a novel high-dimensional time-varying coefficient estimation method, based on high-dimensional Itô diffusion processes. To account for high-dimensional time-varying coefficients, we first estimate local (or instantaneous) coefficients using a time-localized Dantzig selection scheme under a sparsity condition, which results in biased local coefficient estimators due to the regularization. To handle the bias, we propose a debiasing scheme, which provides well-performing unbiased local coefficient estimators. With the unbiased local coefficient estimators, we estimate the integrated coefficient, and to further account for the sparsity of the coefficient process, we apply thresholding schemes. We call this Thresholding dEbiased Dantzig (TED). We establish asymptotic properties of the proposed TED estimator. In the empirical analysis, TED achieves a higher average out-of-sample $R^2$ across assets than benchmark estimators in most periods. Industry-related factors play a central role in explaining asset returns. The estimated integrated coefficients show pronounced time variation associated with firm-specific events and seasonal patterns.
The Dowling geometry $Q_n(Γ)$, where $Γ$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(Γ)$ for various choices of $N$, including subgeometries $Q_m(Γ')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $Γ$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $Γ$ is nontrivial, a complex dependence on $Γ$ emerges, even when $N=M(K_4)$.
Quantum computers based on cold-atom arrays offer long-lived qubits with programmable connectivity, yet their progress toward fault-tolerant operation is limited by the relatively low fidelity of site-selective local control. We introduce an artificial-intelligence (AI) framework that overcomes this limitation. Trained on atom-laser dynamics, a deep neural network autonomously designs composite pulses that improve local control fidelities tenfold while remaining compatible with existing control hardware. We further demonstrate the robustness of these pulses against optical aberrations and beam misalignment. This approach establishes AI-trained pulse compilation for high-fidelity qubit control and can be readily extended to other atom-like platforms, such as trapped ions and solid-state color centers.
Baker and Bowler showed that the Grassmannian can be defined over a tract, a field-like structure generalizing both partial fields and hyperfields. This notion unifies theories of matroids over partial fields, valuated matroids, and oriented matroids. We extend Baker--Bowler theory to the Lagrangian Grassmannian which is the set of maximal isotropic subspaces in a $2n$-dimensional symplectic vector space. By Boege et al., the Lagrangian Grassmannian is parameterized as a subset of the projective space of dimension $2^{n-2}(4+\binom{n}{2})-1$ and its image is cut out by certain quadrics. We simplify a list of quadrics so that these are apparently induced by the Laplace expansions only concerning principal and almost-principal minors of a symmetric matrix. From the idea that the strong basis exchange axiom of matroids captures the combinatorial essence of the Grassmann--Plücker relations, we define matroid-like objects, called antisymmetric matroids, derived from the quadrics for the Lagrangian Grassmannian. We also provide a cryptomorphic definition in terms of circuits capturing the orthogonality and maximality of a Lagrangian subspace. We define antisymmetric matroids over tracts in two equivalent ways, which generalize both BB theory and the parameterization of the Lagrangian Grassmannian. It provides a new perspective on the Lagrangian Grassmannian over hyperfields such as the tropical hyperfield and the sign hyperfield. Our proof involves a homotopy theorem for graphs associated with antisymmetric matroids, which generalizes Maurer's homotopy theorem for matroids. We also prove that if a point in the projective space satisfies the $3$-/$4$-term quadratic relations for the Lagrangian Grassmannian and its supports form the bases of an antisymmetric matroid, then it satisfies all quadratic relations, a result motivated by the earlier work of Tutte for matroids and the Grassmannian.
This paper introduces a novel process for both factor and idiosyncratic volatility matrices whose eigenvalues follow the vector auto-regressive (VAR) model. We call it the factor and idiosyncratic VAR (FIVAR) model. The FIVAR model accounts for the dynamics of the factor and idiosyncratic volatilities and includes many parameters. In addition, many empirical studies have shown that high-frequency stock returns and volatilities often exhibit heavy tails. To handle these two problems simultaneously, we propose a penalized optimization procedure with a truncation scheme for parameter estimation. We apply the proposed parameter estimation procedure to predicting large volatility matrices and establish its asymptotic properties.
In this paper, we develop a novel high-dimensional coefficient estimation procedure based on high-frequency data. Unlike usual high-dimensional regression procedures such as LASSO, we additionally handle the heavy-tailedness of high-frequency observations as well as time variations of coefficient processes. Specifically, we employ the Huber loss and a truncation scheme to handle heavy-tailed observations, while $\ell_{1}$-regularization is adopted to overcome the curse of dimensionality. To account for the time-varying coefficient, we estimate local coefficients which are biased due to the $\ell_{1}$-regularization. Thus, when estimating integrated coefficients, we propose a debiasing scheme to enjoy the law of large numbers property and employ a thresholding scheme to further accommodate the sparsity of the coefficients. We call this Robust thrEsholding Debiased LASSO (RED-LASSO) estimator. We show that the RED-LASSO estimator can achieve a near-optimal convergence rate. In the empirical study, we apply the RED-LASSO procedure to the high-dimensional integrated coefficient estimation using high-frequency trading data.
Reinforcement Learning (RL) has proven effective in solving complex decision-making tasks across various domains, but challenges remain in continuous-time settings, particularly when state dynamics are governed by stochastic differential equations (SDEs) with jump components. In this paper, we address this challenge by introducing the Mean-Square Bipower Variation Error (MSBVE) algorithm, which enhances robustness and convergence in scenarios involving significant stochastic noise and jumps. We first revisit the Mean-Square TD Error (MSTDE) algorithm, commonly used in continuous-time RL, and highlight its limitations in handling jumps in state dynamics. The proposed MSBVE algorithm minimizes the mean-square quadratic variation error, offering improved performance over MSTDE in environments characterized by SDEs with jumps. Simulations and formal proofs demonstrate that the MSBVE algorithm reliably estimates the value function in complex settings, surpassing MSTDE's performance when faced with jump processes. These findings underscore the importance of alternative error metrics to improve the resilience and effectiveness of RL algorithms in continuous-time frameworks.
We develop a theory of representations of (discrete) polymatroids over tracts in terms of Plücker coordinates and suitable Plücker relations. As special cases, we recover polymatroids themselves as polymatroid representations over the Krasner hyperfield K and M-convex functions as polymatroid representations over the tropical hyperfield. We introduce and study several useful operations for polymatroid representations, such as translation and refined notions of minors and duality which have better properties than the existing definitions; for example, deletion and contraction become dual operations (up to translation) in our setting. We also prove an idempotency principle which asserts that polymatroids which are not translates of matroids are representable only over tracts that are idempotent in a certain specific sense (in particular -1 = 1). The space of all representations of a polymatroid J, which we call the thin Schubert cell of J, is represented by an algebraic object called {universal tract of J. When we restrict to just the 3-term Plücker relations, we obtain the weak thin Schubert cell, and passing to torus orbits yields the realization space. These are represented by the universal pasture and the foundation of J, respectively. We exhibit a canonical bijection between the universal tract and the universal pasture, which is new even in the case of matroids, and we show that the foundation of a polymatroid is generated by cross ratios. We also describe a (possibly incomplete) list of multiplicative relations between cross ratios. Thin Schubert cells and realization spaces are canonically embedded in certain tori. Over idempotent tracts, we show that thin Schubert cells contain a canonical torus orbit and split naturally as a product of the realization space with this distinguished torus.