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Hierarchical matrix computations have attracted significant attention in the science and engineering community as exploiting data-sparse structures can significantly reduce the computational complexity of many important kernels. One particularly popular option within this class is the Hierarchical Off-Diagonal Low-Rank (HODLR) format. In this paper, we show that the off-diagonal blocks of HODLR matrices that are approximated by low-rank matrices can be represented in low precision without degenerating the quality of the overall approximation (with the error growth bounded by a factor of $2$). We also present an adaptive-precision scheme for constructing and storing HODLR matrices, and we prove that the use of mixed precision does not compromise the numerical stability of the resulting HOLDR matrix--vector product and LU factorization. That is, the resulting error in these computations is not significantly greater than the case where we use one precision (say, double) for constructing and storing the HOLDR matrix. Our analyses further give insight on how one must choose the working precision in HOLDR matrix computations relative to the approximation error in order to not observe the effects of finite precision. Intuitively, when a HOLDR matrix is subject to a high degree of approximation error, subsequent computations can be performed in a lower precision without detriment. We demonstrate the validity of our theoretical results through a range of numerical experiments.

We investigate the likelihood ratio test for a large block-diagonal covariance matrix with an increasing number of blocks under the null hypothesis. While so far the likelihood ratio statistic has only been studied for normal populations, we establish that its asymptotic behavior is invariant under a much larger class of distributions. This implies robustness against model misspecification, which is common in high-dimensional regimes. Demonstrating the flexibility of our approach, we additionally establish asymptotic normality of the log-likelihood ratio test statistic for the equality of many large sample covariance matrices under model uncertainty. For this statistic, a subtle adjustment to the centering term is needed compared to normal case. A simulation study and an analysis of a data set from psychology emphasize the usefulness of our findings.

We compare the properties of the stable rank and intrinsic dimension of real and complex matrices to those of the classical rank. Basic proofs and examples illustrate that the stable rank does not satisfy any of the fundamental rank properties, while the intrinsic dimension satisfies a few. In particular, the stable rank and intrinsic dimension of a submatrix can exceed those of the original matrix; adding a Hermitian positive semi-definite matrix can lower the intrinsic dimension of the sum; and multiplication by a nonsingular matrix can drastically change the stable rank and the intrinsic dimension. We generalize the concept of stable rank to the p-stable in a Schatten p-norm, thereby unifying the concepts of stable rank and intrinsic dimension: The stable rank is the 2-stable rank, while the intrinsic dimension is the 1-stable rank of a Hermitian positive semi-definite matrix. We derive sum and product inequalities for the pth root of the p-stable rank, and show that it is well-conditioned in the norm-wise absolute sense. The conditioning improves if the matrix and the perturbation are Hermitian positive semi-definite.

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

Image Edge detection (ED) is a base task in computer vision. While the performance of the ED algorithm has been improved greatly by introducing CNN-based models, current models still suffer from unsatisfactory precision rates especially when only a low error toleration distance is allowed. Therefore, model architecture for more precise predictions still needs an investigation. On the other hand, the unavoidable noise training data provided by humans would lead to unsatisfactory model predictions even when inputs are edge maps themselves, which also needs improvement. In this paper, more precise ED models are presented with cascaded skipping density blocks (CSDB). Our models obtain state-of-the-art(SOTA) predictions in several datasets, especially in average precision rate (AP), which is confirmed by extensive experiments. Moreover, our models do not include down-sample operations, demonstrating those widely believed operations are not necessary. Also, a novel modification on data augmentation for training is employed, which allows noiseless data to be employed in model training and thus improves the performance of models predicting on edge maps themselves.

Clustering algorithms frequently require the number of clusters to be chosen in advance, but it is usually not clear how to do this. To tackle this challenge when clustering within sequential data, we present a method for estimating the number of clusters when the data is a trajectory of a Block Markov Chain. Block Markov Chains are Markov Chains that exhibit a block structure in their transition matrix. The method considers a matrix that counts the number of transitions between different states within the trajectory, and transforms this into a spectral embedding whose dimension is set via singular value thresholding. The number of clusters is subsequently estimated via density-based clustering of this spectral embedding, an approach inspired by literature on the Stochastic Block Model. By leveraging and augmenting recent results on the spectral concentration of random matrices with Markovian dependence, we show that the method is asymptotically consistent - in spite of the dependencies between the count matrix's entries, and even when the count matrix is sparse. We also present a numerical evaluation of our method, and compare it to alternatives.

Entropy comparison inequalities are obtained for the differential entropy $h(X+Y)$ of the sum of two independent random vectors $X,Y$, when one is replaced by a Gaussian. For identically distributed random vectors $X,Y$, these are closely related to bounds on the entropic doubling constant, which quantifies the entropy increase when adding an independent copy of a random vector to itself. Consequences of both large and small doubling are explored. For the former, lower bounds are deduced on the entropy increase when adding an independent Gaussian, while for the latter, a qualitative stability result for the entropy power inequality is obtained. In the more general case of non-identically distributed random vectors $X,Y$, a Gaussian comparison inequality with interesting implications for channel coding is established: For additive-noise channels with a power constraint, Gaussian codebooks come within a $\frac{{\sf snr}}{3{\sf snr}+2}$ factor of capacity. In the low-SNR regime this improves the half-a-bit additive bound of Zamir and Erez (2004). Analogous results are obtained for additive-noise multiple access channels, and for linear, additive-noise MIMO channels.

Morphic sequences form a natural class of infinite sequences, typically defined as the coding of a fixed point of a morphism. Different morphisms and codings may yield the same morphic sequence. This paper investigates how to prove that two such representations of a morphic sequence by morphisms represent the same sequence. In particular, we focus on the smallest representations of the subsequences of the binary Fibonacci sequence obtained by only taking the even or odd elements. The proofs we give are induction proofs of several properties simultaneously, and are typically found fully automatically by a tool that we developed.

The convergence rate of various first-order optimization algorithms is a pivotal concern within the numerical optimization community, as it directly reflects the efficiency of these algorithms across different optimization problems. Our goal is making a significant step forward in the formal mathematical representation of optimization techniques using the Lean4 theorem prover. We first formalize the gradient for smooth functions and the subgradient for convex functions on a Hilbert space, laying the groundwork for the accurate formalization of algorithmic structures. Then, we extend our contribution by proving several properties of differentiable convex functions that have not yet been formalized in Mathlib. Finally, a comprehensive formalization of these algorithms is presented. These developments are not only noteworthy on their own but also serve as essential precursors to the formalization of a broader spectrum of numerical algorithms and their applications in machine learning as well as many other areas.

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the \textit{typical table}. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two dual variables, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in \( L^{1} \) to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We derive several new results for random contingency tables from our general framework.