A matrix code is a set of m × n matrices over Fq which form a subspace of the vector space of m × n matrices over the finite field Fq. Matrix codes are related to rank distance codes and have an application to linear network coding. Although there are several papers on self-dual matrix codes, less attention was made to linear complementary dual (LCD) matrix codes. In this paper, we propose two construction methods to generate many LCD matrix codes: Maximum Rank Distance (MRD) and building up construction. In particular, we construct MRD LCD matrix [m × m,m] codes over F2 with largest minimum rank weight m for any positive integer m. We also show the existence of LCD [m×n, k] matrix codes over F2 with the largest minimum rank weight min{m, n} for any k ≥ 1 and m, n ≥ k. For odd q, we establish the existence of optimal LCD [m × n, k] matrix codes over Fq under certain conditions, provided m, n ≥ k ≥ 1.
We study the problem of online clustering of data sequences in the multi-armed bandit (MAB) framework under the fixed-confidence setting. There are M arms, each providing i.i.d. samples from a parametric distribution whose parameters are unknown. The M arms form K clusters based on the distance between the true parameters. In the MAB setting, one arm can be sampled at each time. The objective is to estimate the clusters of the arms using as few samples as possible from the arms, subject to an upper bound on the error probability. Our setting allows for: arms within a cluster to have non-identical distributions, vector parameter arms, vector observations, and K ≤ M clusters. We propose and analyze the Average Tracking Bandit Online Clustering (ATBOC) algorithm. For multivariate Gaussian distributed arms, we show that ATBOC is asymptotically order-optimal, i.e., the expected sample complexity grows at most twice as fast as that of lower bound in asymptotic regime (δ → 0). This upper bound on expected sample complexity is also valid for multivariate sub-Gaussian arms. For single-parameter exponential family distributed arms, we show that ATBOC is asymptotically optimal, i.e., the expected sample complexity growth matches the lower bound in the asymptotic regime. We also propose a computationally more efficient alternatives Lower and Upper Confidence Bound based Bandit Online Clustering Algorithm (LUCBBOC), and Bandit Online Clustering-Elimination (BOC-ELIM). We derive the computational complexity of all the proposed algorithms and also compare the per-sample run time through simulations. The LUCBBOC and BOC-ELIM algorithms require lesser per-sample run time compared to ATBOC and achieve comparable performance. All the proposed algorithms are δ-Probably correct, i.e., the error probability of cluster estimate at the stopping time is upper bounded by δ. We validate the asymptotic optimality guarantees through simulations, and present the comparison of our proposed algorithms with other related work through simulations on both synthetic and real-world datasets.
This work studies the Geometric Jensen-Shannon divergence, based on the notion of geometric interpolation of probability measures, in the setting of Gaussian measures on an infinite-dimensional Hilbert space. On the set of all Gaussian measures equivalent to a fixed one, we present a closed form expression for this divergence that directly generalizes the finite-dimensional version. Using the notion of infinite-dimensional Log-Determinant divergences between positive definite unitized trace class operators, we then define a Regularized Geometric Jensen-Shannon divergence that is valid for any pair of Gaussian measures and that recovers the exact Geometric Jensen-Shannon divergence between two equivalent Gaussian measures when the regularization parameter tends to zero. In the Gaussian process setting, we show that the Regularized Geometric Jensen-Shannon divergence between two centered Gaussian processes admits consistent and efficient finite-dimensional approximation, which can readily be employed in numerical algorithms, with dimension-independent sample complexity.
We study the problem of constructing (rho, L)-list-decodable codes C subset of F-q(n) with small q using minimal randomness. The central goal is to generate codes of rate approaching the Elias bound, that is, rate at least 1-h(rho)-O(1/L), using significantly fewer random bits than required by uniformly random linear codes. Prior combinatorial constructions achieve this using O(Ln) random bits via graph-based methods. In this work, we present two new and fully algebraic constructions that match this randomness efficiency while offering greater simplicity and structural transparency. Our first construction, a generalization of the Wozencraft ensemble, achieves the Elias bound with only Ln random bits; its dual achieves the Gilbert-Varshamov bound, and both codes support quasilinear-time encoding. Our second construction uses 2nL random bits and yields a code whose dual also achieves the Elias bound. These dual properties are critical for applications in areas such as cryptography. Our analysis proceeds by designing codes that replicate key local properties of random linear codes, allowing us to invoke known results to deduce list-decodability. As a final contribution, we prove a lower bound showing that any construction relying solely on such local approximation must use at least L(1-R)n log(2)(q) random bits to obtain rate-R codes over an alphabet of size q.
