arXiv subjects

Papri Dey

Publications and source records attributed to Papri Dey.

At least 19 recordsLinked to original sources

arXiv2026

Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships

Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.

stat.ME↗

arXiv2026

Sampled-data Robust Control of Electrically Stimulated Engineered Cell Factories

Closed-loop bioelectronic regulation of engineered secretory cell systems is challenging because electric-field (EF) stimulation acts indirectly through transcription-factor activation, in the presence of delayed, nonlinear, and noisy intracellular dynamics, sparse measurements, and constrained burst-based actuation. We develop a framework for robust closed-loop endocrine regulation in electrically stimulated engineered cell factories, illustrated through extracellular thyroid hormone $T_4$ production in engineered thyroid-like cells. The plant is modeled by a control-oriented ODE formulation combining a reduced mechanistic $T_4$ pathway, an EF-responsive Hill module, and a linear-chain Erlang cascade representing distributed intracellular delay. On this basis, we design a sampled-data adaptive proportional-integral-derivative (PID) controller with derivative filtering, anti-windup, saturation and rate limits, and hysteretic band-locking, together with a robust adaptive extension that accounts for parameter mismatch, sensor noise and bias, actuator mismatch, delay/jitter, and exogenous rhythmic disturbance through a scenario-based risk-aware update. We provide local sampled-data input-to-state stability interpretations for both APID and RAPID, showing that, under standard local Lyapunov and bounded-disturbance conditions, the sampled tracking error is ultimately bounded by a disturbance-dependent constant. In silico experiments demonstrate sustained regulation of extracellular $T_4$ across prescribed targets despite significant uncertainty.

eess.SY↗

arXiv2026

Geometry-Aware Langevin Sampling for Matrix-Valued Graph Learning

Bayesian inference over positive semidefinite (PSD) matrix-valued parameters arises in structured covariance estimation, graph-Laplacian precision models, and multi-output graph learning, but Euclidean proposals often mix poorly near the cone boundary. We propose \ConeMALA, a geometry-aware Metropolis-adjusted Langevin algorithm whose proposal geometry is induced by the model's log-determinant structure. For a PSD-weighted graph with edge kernels $W_e\succeq 0$, block Laplacian $L(W)$ , and stabilizer $R\succ 0$, the lifted precision matrix $X(W)=L(W)+R\in \mathbb S_{++}^{md}$ defines the log-determinant energy $\Phi(W)=-\log\det X(W).$ We show that the Hessian of $\Phi$ is the pullback of the affine-invariant SPD metric under the map $W\mapsto X(W)$, yielding explicit intrinsic Langevin proposals with Metropolis-Hastings correction using the closed-form SPD exponential-map Jacobian. We validate the metric on rank-one PSD edge perturbations for $d=5$, obtaining essentially exact agreement between analytic curvature scores and finite-difference curvatures. In intrinsic SPD posterior and matrix-valued graph Gaussian experiments, \ConeMALA achieves stable multichain diagnostics and substantially higher ESS/sec than Euclidean MALA and generic RMALA, while a PDHMC-like finite-difference baseline is accurate but computationally prohibitive at larger graph sizes. These results show that pullback log-determinant geometry provides a practical route to uncertainty quantification in PSD-constrained graph learning.

math.OC↗

arXiv2025

Phylogenetics in a warm place: computational aspects of the Tropical Grassmannian

Phylogenetic trees provide a fundamental representation of evolutionary relationships, yet the combinatorial explosion of possible tree topologies renders inference computationally challenging. Classical approaches to characterizing tree space, such as the Billera-Holmes-Vogtmann (BHV) space, offer elegant geometric structure but suffer from statistical and computational limitations. An alternative perspective arises from tropical geometry, the tropical Grassmannian tropGr(2,n), introduced by Speyer and Sturmfels, which coincides with phylogenetic tree space. In this paper, we review the structure of the tropical Grassmannian and present algorithmic methods for its computational study, including procedures for sampling from the tropical Grassmannian. Our aim is to make these concepts accessible to evolutionary biologists and computational scientists, and to motivate new research directions at the interface of algebraic geometry and phylogenetic inference.

q-bio.PE↗

arXiv2025

$K-$Lorentzian Polynomials, Semipositive Cones, and Cone-Stable EVI Systems

Lorentzian and completely log-concave polynomials have recently emerged as a unifying framework for negative dependence, log-concavity, and convexity in combinatorics and probability. We extend this theory to variational analysis and cone-constrained dynamics by studying $K$-Lorentzian and $K$-completely log-concave polynomials over a proper convex cone $K\subset\mathbb{R}^n$. For a $K$-Lorentzian form $f$ and $v\in\operatorname{int}K$, we define an open cone $K^\circ(f,v)$ and a closed cone $K(f,v)$ via directional derivatives along $v$, recovering the usual hyperbolicity cone when $f$ is hyperbolic. We prove that $K^\circ(f,v)$ is a proper cone and equals $\operatorname{int}K(f,v)$. If $f$ is $K(f,v)$-Lorentzian, then $K(f,v)$ is convex and maximal among convex cones on which $f$ is Lorentzian. Using the Rayleigh matrix $M_f(x)=\nabla f(x)\nabla f(x)^T - f(x)\nabla^2 f(x)$, we obtain cone-restricted Rayleigh inequalities and show that two-direction Rayleigh inequalities on $K$ are equivalent to an acuteness condition for the bilinear form $v^T M_f(x) w$. This yields a cone-restricted negative-dependence interpretation linking the curvature of $\log f$ to covariance properties of associated Gibbs measures. For determinantal generating polynomials, we identify the intersection of the hyperbolicity cone with the nonnegative orthant as the classical semipositive cone, and we extend this construction to general proper cones via $K$-semipositive cones. Finally, for linear evolution variational inequality (LEVI) systems, we show that if $q(x)=x^T A x$ is (strictly) $K$-Lorentzian, then $A$ is (strictly) $K$-copositive and yields Lyapunov (semi-)stability on $K$, giving new Lyapunov criteria for cone-constrained dynamics.

math.OC↗

arXiv2025

$\K$-Lorentzian and $\K$-CLC Polynomials in Stability Analysis

We study the class of $\K$-Lorentzian polynomials, a generalization of the distinguished class of Lorentzian polynomials. As shown in \cite{GPlorentzian}, the set of $\K$-Lorentzian polynomials is equivalent to the set of $\K$-completely log-concave (aka $\K$-CLC) forms. Throughout this paper, we interchangeably use the terms $\K$-Lorentzian polynomials for the homogeneous setting and $\K$-CLC polynomials for the non-homogeneous setting. By introducing an alternative definition of $\K$-CLC polynomials through univariate restrictions, we establish that any strictly $\K$-CLC polynomial of degree $d \leq 4$ is Hurwitz-stable polynomial over $\K$. Additionally, we characterize the conditions under which a strictly $\K$-CLC of degree $d \geq 5$ is Hurwitz-stable over $\K$. Furthermore, we associate the largest possible proper cone, denoted by $\K(f,v)$, with a given $\K$-Lorentzian polynomial $f$ in the direction $v \in \inter \K$. Finally, we investigate applications of $\K$-CLC polynomials in the stability analysis of evolution variational inequalities (EVI) dynamical systems governed by differential equations and inequality constraints.

math.DS↗

arXiv2024

$\mathcal{K}$-Lorentzian Polynomials

Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone $\mathcal{K}$ we find a connection between $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone $\mathcal{K}$ varies, even the set of quadratic $\mathcal{K}$-Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, $\mathcal{K}$-Lorentzian and $\mathcal{K}$-completely log-concave polynomials coincide.

math.AG↗

arXiv2022

Polynomials with Lorentzian Signature, and Computing Permanents via Hyperbolic Programming

We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic polynomials and conic stable polynomials belong to this class, and the set of polynomials with Lorentzian signature is closed. Finally, we develop a method for computing permanents of nonsingular matrices which belong to a class that includes nonsingular $k$-locally singular matrices via hyperbolic programming.

math.AG↗

arXiv2021

Bit Complexity of Jordan Normal Form and Spectral Factorization

We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an $\epsilon-$approximation to the Jordan Normal form of an integer matrix with $a-$bit entries, where $\omega$ is the exponent of matrix multiplication. (2) An $\tilde{O}(n^6d^6a+n^4d^4a^2+n^3d^3\log(1/\epsilon))$ time algorithm for $\epsilon$-approximately computing the spectral factorization $P(x)=Q^*(x)Q(x)$ of a given monic $n\times n$ rational matrix polynomial of degree $2d$ with rational $a-$bit coefficients having $a-$bit common denominators, which satisfies $P(x)\succeq 0$ for all real $x$. The first algorithm is used as a subroutine in the second one. Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best running time was an unspecified polynomial in $n$ of degree at least twelve \cite{cai1994computing}. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself.

cs.DS↗

arXiv2021

Real degeneracy loci of matrices and phase retrieval

Let ${\mathcal A} = \{A_{1},\dots,A_{r}\}$ be a collection of linear operators on ${\mathbb R}^m$. The degeneracy locus of ${\mathcal A}$ is defined as the set of points $x \in {\mathbb P}^{m-1}$ for which rank$([A_1 x \ \dots \ A_{r} x]) \\ \leq m-1$. Motivated by results in phase retrieval we study degeneracy loci of four linear operators on ${\mathbb R}^3$ and prove that the degeneracy locus consists of 6 real points obtained by intersecting four real lines if and only if the collection of matrices lies in the linear span of four fixed rank one operators. We also relate such {\em quadrilateral configurations} to the singularity locus of the corresponding Cayley cubic symmetroid. More generally, we show that if $A_i , i = 1, \dots, m + 1$ are in the linear span of $m + 1$ fixed rank-one matrices, the degeneracy locus determines a {\em generalized Desargues configuration} which corresponds to a Sylvester spectrahedron.

math.AG↗

arXiv2020

Principal Matrices of Numerical Semigroups

Principal matrices of a numerical semigroup of embedding dimension n are special types of $n \times n$ matrices over integers of rank $\leq n - 1$. We show that such matrices and even the pseudo principal matrices of size n must have rank $\geq \frac{n}{2}$ regardless of the embedding dimension. We give structure theorems for pseudo principal matrices for which at least one $n - 1 \times n - 1$ principal minor vanish and thereby characterize the semigroups in embedding dimensions $4$ and $5$ in terms of their principal matrices. When the pseudo principal matrix is of rank $n - 1$, we give a sufficient condition for it to be principal.

math.AC↗

arXiv2020

The $4 \times 4$ orthostochastic variety

Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the real variety were previously known only for $3 \times 3$ matrices. We study the real variety of $4 \times 4$ orthostochastic matrices, and find a minimal defining set of equations consisting of 6 quintics and 3 octics. The techniques used here involve a wide range of both symbolic and computational methods, in computer algebra and numerical algebraic geometry.

math.AG↗

arXiv2019

Conic stability of polynomials and positive maps

Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f \in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the non-negative orthant, then $K$-stability specializes to the usual notion of stability of polynomials. We study conditions and certificates for the $K$-stability of a given polynomial $f$, especially for the case of determinantal polynomials as well as for quadratic polynomials. A particular focus is on psd-stability. For cones $K$ with a spectrahedral representation, we construct a semidefinite feasibility problem, which, in the case of feasibility, certifies $K$-stability of $f$. This reduction to a semidefinite problem builds upon techniques from the connection of containment of spectrahedra and positive maps. In the case of psd-stability, if the criterion is satisfied, we can explicitly construct a determinantal representation of the given polynomial. We also show that under certain conditions, for a $K$-stable polynomial $f$, the criterion is at least fulfilled for some scaled version of $K$.

math.AG↗

arXiv2019

Computing symmetric determinantal representations

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are geared towards speed and robustness, employing linear algebra and numerical algebraic geometry, without genericity assumptions on the polynomials.

math.AG↗

arXiv2018

Testing hyperbolicity of real polynomials

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.

math.AG↗

arXiv2018

Coordinate-wise Powers of Algebraic Varieties

We introduce and study coordinate-wise powers of subvarieties of $\mathbb{P}^n$, i.e. varieties arising from raising all points in a given subvariety of $\mathbb{P}^n$ to the $r$-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of $\mathbb{P}^n$ under the quotient of $\mathbb{P}^n$ by the action of the finite group $\mathbb{Z}_r^{n+1}$. We determine the degree of coordinate-wise powers and study their defining equations, particularly for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.

math.AG↗

arXiv2017

Definite Determinantal Representations of Multivariate Polynomials

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree $d$. Then we propose a heuristic method to obtain a monic symmetric determinantal representation of a multivariate polynomial of degree $d$.

math.OC↗

arXiv2017

Definite Determinantal Representations via Orthostochastic Matrices

Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a determinantal representation. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree $d$. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. We also characterize the range set of vector coefficients of a certain type of determinantal bivariate polynomials.

math.OC↗