Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers $k,\ell$, large enough dicritical digraphs with dichromatic number $k$ contain an orientation of a cycle with at least $\ell$ vertices; (ii) there are functions $f,g$ such that for every subdivision $F^*$ of a digraph $F$, digraphs with directed girth at least $f(F^*)$ and dichromatic number at least $g(F)$ contain a subdivision of $F^*$, and if $F$ is a tree, then $g(F)=|V(F)|$; (iii) there is a function $f$ such that for every subdivision $F^*$ of $TT_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f(F^*)$ and minimum out-degree at least $2$ contain $F^*$ as a subdivision.
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Asymptotic expansion of a Hurst index estimator for a stochastic differential equation driven by fBm
We study the asymptotic properties of an estimator of Hurst parameter of a stochastic differential equation driven by a fractional Brownian motion with $H > 1/2$. Utilizing the theory of asymptotic expansion of Skorohod integrals introduced by Nualart and Yoshida [NY19], we derive an asymptotic expansion formula of the distribution of the estimator. As an corollary, we also obtain a mixed central limit theorem for the statistic, indicating that the rate of convergence is $n^{-\frac12}$, which improves the results in the previous literature. To handle second-order quadratic variations appearing in the estimator, a theory of exponent has been developed based on weighted graphs to estimate asymptotic orders of norms of functionals involved.
Tools from optimal transport (OT) theory have recently been used to define a notion of quantile function for directional data. In practice, regularization is mandatory for applications that require out-of-sample estimates. To this end, we introduce a regularized estimator built from entropic optimal transport, by extending the definition of the entropic map to the spherical setting. We propose a stochastic algorithm to directly solve a continuous OT problem between the uniform distribution and a target distribution, by expanding Kantorovich potentials in the basis of spherical harmonics. In addition, we define the directional Monge-Kantorovich depth, a companion concept for OT-based quantiles. We show that it benefits from desirable properties related to Liu-Zuo-Serfling axioms for the statistical analysis of directional data. Building on our regularized estimators, we illustrate the benefits of our methodology for data analysis.
Fitting's Heyting-valued logic and Heyting-valued modal logic have already been studied from an algebraic viewpoint. In addition to algebraic axiomatizations with the completeness of Fitting's Heyting-valued logic and Heyting-valued modal logic, both topological and coalgebraic dualities have also been developed for algebras of Fitting's Heyting-valued modal logic. Bitopological methods have recently been employed to investigate duality for Fitting's Heyting-valued logic. However, the concepts of bitopology and biVietoris coalgebras are conspicuously absent from the development of dualities for Fitting's many-valued modal logic. With this study, we try to bridge that gap. We develop a bitopological duality for algebras of Fitting's Heyting-valued modal logic. We construct a bi-Vietoris functor on the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued ($\mathcal{L}$ is a Heyting algebra) pairwise Boolean spaces. Finally, we obtain a dual equivalence between categories of biVietoris coalgebras and algebras of Fitting's Heyting-valued modal logic. As a result, we conclude that Fitting's many-valued modal logic is sound and complete with respect to the coalgebras of a biVietoris functor. We discuss the application of this coalgebraic approach to bitopological duality.
The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments \cite{DrMoZa3}, where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in \cite{DrMoZa3} to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.
Machine Learning (ML) in low-data settings remains an underappreciated yet crucial problem. Hence, data augmentation methods to increase the sample size of datasets needed for ML are key to unlocking the transformative potential of ML in data-deprived regions and domains. Unfortunately, the limited training set constrains traditional tabular synthetic data generators in their ability to generate a large and diverse augmented dataset needed for ML tasks. To address this challenge, we introduce CLLM, which leverages the prior knowledge of Large Language Models (LLMs) for data augmentation in the low-data regime. However, not all the data generated by LLMs will improve downstream utility, as for any generative model. Consequently, we introduce a principled curation mechanism, leveraging learning dynamics, coupled with confidence and uncertainty metrics, to obtain a high-quality dataset. Empirically, on multiple real-world datasets, we demonstrate the superior performance of CLLM in the low-data regime compared to conventional generators. Additionally, we provide insights into the LLM generation and curation mechanism, shedding light on the features that enable them to output high-quality augmented datasets.
The purpose of this technical note is to summarize the relationship between the marginal variance and correlation length of a Gaussian random field with Mat\'ern covariance and the coefficients of the corresponding partial-differential-equation (PDE)-based precision operator.
We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.
Bayesian inference gets its name from *Bayes's theorem*, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes's theorem. Many hypotheses of scientific interest are *composite hypotheses*, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses by *marginalization* over auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes's theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces -- both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification.
We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy.
This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We rigorously analyze the diagonal empirical Fisher information matrix (FIM) in Adam, clarifying all detailed approximations and advocating for the use of log probability functions as loss, which should be based on discrete distributions, due to the limitations of empirical FIM. Our analysis uncovers flaws in the original Adam algorithm, leading to proposed corrections such as enhanced momentum calculations, adjusted bias corrections, adaptive epsilon, and gradient clipping. We refine the weight decay term based on our theoretical framework. Our modified algorithm, Fisher Adam (FAdam), demonstrates superior performance across diverse domains including LLM, ASR, and VQ-VAE, achieving state-of-the-art results in ASR.
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