Artan Sheshmani

Artan Sheshmani, Yizhuang You, Wenbo Fu et al.

Following the earlier formalism of the categorical representation learning (arXiv:2103.14770) by the first two authors, we discuss the construction of the "RG-flow based categorifier". Borrowing ideas from theory of renormalization group flows (RG) in quantum field theory, holographic duality, and hyperbolic geometry, and mixing them with neural ODE's, we construct a new algorithmic natural language processing (NLP) architecture, called the RG-flow categorifier or for short the RG categorifier, which is capable of data classification and generation in all layers. We apply our algorithmic platform to biomedical data sets and show its performance in the field of sequence-to-function mapping. In particular we apply the RG categorifier to particular genomic sequences of flu viruses and show how our technology is capable of extracting the information from given genomic sequences, find their hidden symmetries and dominant features, classify them and use the trained data to make stochastic prediction of new plausible generated sequences associated with new set of viruses which could avoid the human immune system. The content of the current article is part of the recent US patent application submitted by first two authors (U.S. Patent Application No.: 63/313.504).

Yi-Shuai Niu, Artan Sheshmani, Shing-Tung Yau

We propose Yau's Affine Normal Descent (YAND), a geometric framework for smooth unconstrained optimization in which search directions are defined by the equi-affine normal of level-set hypersurfaces. The resulting directions are invariant under volume-preserving affine transformations and intrinsically adapt to anisotropic curvature. Using the analytic representation of the affine normal from affine differential geometry, we establish its equivalence with the classical slice-centroid construction under convexity. For strictly convex quadratic objectives, affine-normal directions are collinear with Newton directions, implying one-step convergence under exact line search. For general smooth (possibly nonconvex) objectives, we characterize precisely when affine-normal directions yield strict descent and develop a line-search-based YAND. We establish global convergence under standard smoothness assumptions, linear convergence under strong convexity and Polyak-Lojasiewicz conditions, and quadratic local convergence near nondegenerate minimizers. We further show that affine-normal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations. Numerical experiments illustrate the geometric behavior of the method and its robustness under strong anisotropic scaling.

Ya-Juan Wang, Yi-Shuai Niu, Artan Sheshmani et al.

Unrestricted mean-variance-skewness-kurtosis portfolio optimization can capture asymmetry and tail risk, but sample-moment formulations become computationally impractical when the asset universe is large: they produce dense nonconvex quartic objectives with prohibitive coskewness and cokurtosis tensors and anisotropic, ill-conditioned level sets. We develop a structure-exploiting algorithm based on Yau's affine-normal descent that follows affine-normal directions of the current level set while working directly with the return matrix. The method avoids explicit higher-order tensors and exploits the quartic structure for exact sample oracles, derivative evaluation, and exact line search. We also provide theory for the reduced simplex formulation, including regularity and convexity conditions that separate data-map geometry from investor preference coefficients. Computational results show a clear implementation split: a direct configuration is effective on the standard small benchmark, whereas a preconditioned conjugate-gradient configuration with stall recovery becomes the preferred large-scale implementation by the upper end of the hundreds and remains competitive as the asset universe moves into the thousands. On a 5-minute A-share panel with 5,440 stocks, the method makes direct full-universe comparisons with exact mean-variance portfolios feasible and shows on the baseline split that the incremental value of higher moments is strongest at moderate return targets.

Yi-Shuai Niu, Artan Sheshmani, Shing-Tung Yau

Affine normal directions provide intrinsic affine-invariant descent directions derived from the geometry of level sets. Their practical use, however, has long been hindered by the need to evaluate third-order derivatives and invert tangent Hessians, which becomes computationally prohibitive in high dimensions. In this paper, we show that affine normal computation admits an exact reduction to second-order structure: the classical third-order contraction term is precisely the gradient of the log-determinant of the tangent Hessian. This identity replaces explicit third-order tensor contraction by a matrix-free formulation based on tangent linear solves, Hessian-vector products, and log-determinant gradient evaluation. Building on this reduction, we develop exact and stochastic matrix-free procedures for affine normal evaluation. For sparse polynomial objectives, the algebraic closure of derivatives further yields efficient sparse kernels for gradients, Hessian-vector products, and directional third-order contractions, leading to scalable implementations whose cost is governed by the sparsity structure of the polynomial representation. We establish end-to-end complexity bounds showing near-linear scaling with respect to the relevant sparsity scale under fixed stochastic and Krylov budgets. Numerical experiments confirm that the proposed MF-LogDet formulation reproduces the original autodifferentiation-based affine normal direction to near machine precision, delivers substantial runtime improvements in moderate and high dimensions, and exhibits empirical near-linear scaling in both dimension and sparsity. These results provide a practical computational route for affine normal evaluation and reveal a new connection between affine differential geometry, log-determinant curvature, and large-scale structured optimization.

Artan Sheshmani, Yizhuang You

We provide a construction for categorical representation learning and introduce the foundations of "$\textit{categorifier}$". The central theme in representation learning is the idea of $\textbf{everything to vector}$. Every object in a dataset $\mathcal{S}$ can be represented as a vector in $\mathbb{R}^n$ by an $\textit{encoding map}$ $E: \mathcal{O}bj(\mathcal{S})\to\mathbb{R}^n$. More importantly, every morphism can be represented as a matrix $E: \mathcal{H}om(\mathcal{S})\to\mathbb{R}^{n}_{n}$. The encoding map $E$ is generally modeled by a $\textit{deep neural network}$. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a $\textit{set-theoretic}$ approach. The goal of the current article is to promote the representation learning to a new level via a $\textit{category-theoretic}$ approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).