arXiv subjects

Saugata Basu

Publications and source records attributed to Saugata Basu.

At least 19 recordsLinked to original sources

arXiv2026

Solving polynomial inequalities over spaces of convex sets and applications

We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let $\mathcal B$ be a family of subsets containing $\emptyset$ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in $\mathcal B$, then every coordinate set of the smallest solution lies in $\mathcal B$; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra---where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes---then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For \[ V=U\oplus\bigoplus_{i=1}^{k}W_i,\qquad \dim W_i=1,\qquad \Lambda=\bigcup_{i=1}^{k}(U+W_i), \] we prove that the lamination hull $G_{\Lambda}^{(\infty)}(S)$ of every finite $S\subset V$ is semi-algebraic and effectively computable by a quantifier-free formula over the reals.

math.CO↗

arXiv2026

Essential Simplices Dominate in Harmonic Representatives of One-Dimensional Persistent Classes

Persistent homology summarizes the birth and death of topological features, but it does not by itself specify where a feature is located in the underlying complex. Harmonic persistent homology addresses this by assigning canonical harmonic cycle representatives to bars. In earlier work, Basu and Cox showed that harmonic representatives of simple bars maximize the total relative weight placed on essential simplices, the simplices that are forced to appear in representatives of the corresponding class. In this paper we prove that, for generic one-dimensional bars, this preference is stronger than an aggregate maximization statement. Every essential edge has strictly larger coefficient, in absolute value, than every non-essential edge in the harmonic representative, and the absolute values of the coefficients of all essential edges are equal. The key argument is a finite-dimensional variational characterization of the harmonic representative as a minimum-norm chain with prescribed boundary, combined with an elementary graph-theoretic cut argument. We then prove that the result is special to dimension one. In higher dimensions, the analogous coefficient-wise dominance statement fails. We give examples to show that harmonic representatives can place larger coefficients on non-essential higher-dimensional simplices than on essential ones. These results clarify both the power and the limitations of using harmonic representatives to assign geometric significance to simplices in persistent homology.

math.AT↗

arXiv2025

Thom-Milnor bounds for smooth manifolds

We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends previous results in the Euclidean setting by Lerario and Stecconi [J. Singul., 2021]. As a key step, we generalize the Thom-Milnor bound to polynomial maps on a nonsingular real algebraic variety, improving the dependence on the degree. Finally, inspired by the work of B\"{u}rgisser, Cucker and Tonelli-Cueto [Found. Comput. Math., 2020], we introduce a condition number for families of functions. Using this we extend existing bounds due to Basu, Pollack and Roy [Proc. Amer. Math. Soc., 2004], for the Betti numbers of semialgebraic sets described by closed conditions to what we call closed semialgebraic type sets, namely sets defined by closed inequalities involving smooth functions.

math.AG↗

arXiv2025

On the convergence of critical points on real algebraic sets and applications to optimization

Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an infinitesimal perturbation $V_{\xi} = \zero(\{P_1-\xi_1,\ldots,P_s-\xi_s\},\R\la \xi \ra^n)$. Our main motivation is to understand the limiting behavior of local minimizers of the log-barrier function (and central paths) in polynomial optimization, whose existence plays a fundamental role, in theory and practice, for modern interior point methods. We establish different sets of conditions that ensure existence, finiteness, boundedness, and non-degeneracy of critical points of $F$ on $V_{\xi}$, respectively. These lead to new conditions for the existence, convergence, and smoothness of central paths of polynomial optimization and its extension to non-linear optimization problems involving definable sets and functions in an o-minimal structure. In particular, for non-linear programs defined by real globally analytic functions, our extension provides a stronger form of the convergence result obtained by Drummond and Peterzil.

math.AG↗

arXiv2024

Bounds on the realizations of zero-nonzero patterns and sign conditions of polynomials restricted to varieties and applications

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$ of affine or projective space. The bounds depend only on $\mathrm{card}(\mathcal P)$ and the degrees of the polynomials in $\mathcal P$, together with $\mathrm{deg}(V)$ and $\dim(V)$, and not on the dimension of the space in which $V$ is embedded. This feature is particularly useful when $V$ has small intrinsic dimension but is presented in a very high-dimensional ambient space. We describe several applications. First, we extend existing results on bounding the $\varepsilon$-entropy of real algebraic varieties. Second, we derive lower bounds (in terms of the number of connected components) for membership testing in semi-algebraic sets in the algebraic computation tree model. Finally, motivated by quantum complexity theory, we introduce additive and multiplicative notions of \emph{relative rank} in finite-dimensional vector spaces and algebras with respect to a fixed algebraic subset, generalizing the classical notion of tensor rank. We prove a general lower bound on the maximum relative rank of finite subsets with respect to algebraic sets of bounded degree and dimension that is again independent of the ambient dimension. As an illustration, we obtain a quantum analogue of Shannon's classical lower bound: almost all Boolean functions require classical circuits of size $\Omega(2^n/n)$, even in the presence of a quantum oracle specified by an algebraic subset of fixed degree and dimension.

math.CO↗

arXiv2024

Cohomological VC-density: Bounds and Applications

The concept of Vapnik-Chervonenkis (VC) density is pivotal across various mathematical fields, including discrete geometry, probability theory and model theory. In this paper, we introduce a topological generalization of VC-density. Let $Y$ be a topological space and $\mathcal{X}$ a family of closed subspaces of $Y$. For each $p \geq 0$, we define a number, $\mathrm{vcd}^{p}_{\mathcal{X}}$, which we refer to as the degree $p$ VC-density of the family $\mathcal{X}$. The classical notion of VC-density within this topological framework can be recovered by setting $p=0$. Our definition of degree $p$ VC-density extends to higher orders as well. For $p \geq 0$, $q \geq 1$, we define the degree $p$, order $q$ VC density $\mathrm{vcd}^{p,q}_{\mathcal{X}}$ of $\mathcal{X}$, which recovers Shelah's notion of higher order VC-density for $q$-dependent families when $p=0$. Our definition introduces a completely new notion when $p > 0$. We examine the properties of $\mathrm{vcd}_{\mathcal{X}}^p$ (as well as $\mathrm{vcd}^{p,q}_{\mathcal{X}}$) when the families $\mathcal{X}$ are definable in structures with some underlying topology (for instance, the Euclidean topology for o-minimal structures over $\mathbb{R}$, the analytic topology over $\mathbb{C}$, or the \'{e}tale site for schemes over arbitrary algebraically closed fields). Our main result establishes that in any model of these theories \[ \mathrm{vcd}_{\mathcal{X}}^p \leq (p+1) \dim X, \] and more generally for any $q \geq 1$ \[ \mathrm{vcd}^{p,q}_{\mathcal{X}} \leq (p+q) \dim X. \] We give examples to show that our bounds are optimal. We also present combinatorial applications of our higher-degree VC-density bounds, deriving higher degree topological analogs of well-known results such as the existence of $\varepsilon$-nets and the fractional Helly theorem.

math.LO↗

arXiv2024

Complexity and speed of semi-algebraic multi-persistence

Let $\mathrm{R}$ be a real closed field, $S \subset \mathrm{R}^n$ a closed and bounded semi-algebraic set, and $\mathbf{f}=(f_1,\ldots,f_p):S \rightarrow \mathrm{R}^p$ a continuous semi-algebraic map inducing a $p$-parameter semi-algebraic filtration by sublevel sets. We introduce a barcode invariant for such filtrations that directly extends the classical ($p=1$) barcode. After scaling of the parameter space, in each homological degree $\ell$ the invariant is encoded by a $\mathbb{Z}_{\ge 0}$-valued function \[ \mu_\ell(S,\mathbf{f}):\ \Big(({-}1,1)^p\times(({-}1,1)^p \cup\{(1,\ldots,1)\}) \Big)\ \cap\ \{(\mathbf a,\mathbf b)\mid \mathbf a\preceq \mathbf b\} \ \longrightarrow\ \mathbb{Z}_{\ge 0}, \] where $\preceq$ denotes the product order on $\mathrm{R}^p$. We prove that $\mu_\ell(S,\mathbf{f})$ is semi-algebraically constructible and establish a singly exponential upper bound on its description complexity. Moreover, we give a singly exponential-time algorithm to compute $\mu_\ell(S,\mathbf{f})$, extending to arbitrary $p$ the corresponding result for $p=1$ by Basu and Karisani. Finally, for semi-algebraic filtrations of bounded description complexity we bound the number of equivalence classes of finite poset modules realizable in this way, yielding a tight analogue of "speed" bounds for algebraically defined graph classes.

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arXiv2023

Equivariance in Approximation by Compact Sets

We adapt a construction of Gabrielov and Vorobjov for use in the symmetric case. Gabrielov and Vorobjov had developed a means by which one may replace an arbitrary set $S$ definable in some o-minimal expansion of $\mathbb{R}$ with a compact set $T$. $T$ is constructed in such a way that for a given $m>0$ we have epimorphisms from the first $m$ homotopy and homology groups of $T$ to those of $S$. If $S$ is defined by a boolean combination of statements $h(x)=0$ and $h(x)>0$ for various $h$ in some finite collection of definable continuous functions, one may choose $T$ so that these maps are isomorphisms for $0\leq k\leq m-1$. In this case, $T$ is also defined by functions closely related to those defining $S$. In this paper we study sets $S$ symmetric under the action of some finite reflection group $G$. One may see that in the original construction, if $S$ is defined by functions symmetric relative to the action of $G$, then $T$ will be as well. We show that there is an equivariant map $T\rightarrow S$ inducing the aforementioned epimorphisms and isomorphisms of homotopy and homology groups. We use this result to strengthen theorems of Basu and Riener concerning the multiplicities of Specht modules in the isotypic decomposition of the cohomology spaces of sets defined by polynomials symmetric relative to $\mathfrak{S}_n$.

math.AG↗

arXiv2023

Probing omics data via harmonic persistent homology

Identifying molecular signatures from complex disease patients with underlying symptomatic similarities is a significant challenge in the analysis of high dimensional multi-omics data. Topological data analysis (TDA) provides a way of extracting such information from the geometric structure of the data and identifying multiway higher-order relationships. Here, we propose an application of Harmonic persistent homology, which overcomes the limitations of ambiguous assignment of the topological information to the original elements in a representative topological cycle from the data. When applied to multi-omics data, this leads to the discovery of hidden patterns highlighting the relationships between different omic profiles, while allowing for common tasks in multi-omics analyses, such as disease subtyping, and most importantly biomarker identification for similar latent biological pathways that are associated with complex diseases. Our experiments on multiple cancer data show that harmonic persistent homology effectively dissects multi-omics data to identify biomarkers by detecting representative cycles predictive of disease subtypes.

q-bio.GN↗

arXiv2023

Quantum Analog of Shannon's Lower Bound Theorem

Shannon proved that almost all Boolean functions require a circuit of size $\Theta(2^n/n)$. We prove a quantum analog of this classical result. Unlike in the classical case the number of quantum circuits of any fixed size that we allow is uncountably infinite. Our main tool is a classical result in real algebraic geometry bounding the number of realizable sign conditions of any finite set of real polynomials in many variables.

quant-ph↗

arXiv2023

Towards quantum-enabled cell-centric therapeutics

In recent years, there has been tremendous progress in the development of quantum computing hardware, algorithms and services leading to the expectation that in the near future quantum computers will be capable of performing simulations for natural science applications, operations research, and machine learning at scales mostly inaccessible to classical computers. Whereas the impact of quantum computing has already started to be recognized in fields such as cryptanalysis, natural science simulations, and optimization among others, very little is known about the full potential of quantum computing simulations and machine learning in the realm of healthcare and life science (HCLS). Herein, we discuss the transformational changes we expect from the use of quantum computation for HCLS research, more specifically in the field of cell-centric therapeutics. Moreover, we identify and elaborate open problems in cell engineering, tissue modeling, perturbation modeling, and bio-topology while discussing candidate quantum algorithms for research on these topics and their potential advantages over classical computational approaches.

quant-ph↗

arXiv2023

On the complexity of analyticity in semi-definite optimization

It is well-known that the central path of semi-definite optimization, unlike linear optimization, has no analytic extension to $\mu = 0$ in the absence of the strict complementarity condition. In this paper, we show the existence of a positive integer $\rho$ by which the reparametrization $\mu \mapsto \mu^{\rho}$ recovers the analyticity of the central path at $\mu = 0$. We investigate the complexity of computing $\rho$ using algorithmic real algebraic geometry and the theory of complex algebraic curves. We prove that the optimal $\rho$ is bounded by $2^{O(m^2+n^2m+n^4)}$, where $n$ is the matrix size and $m$ is the number of affine constraints. Our approach leads to a symbolic algorithm, based on the Newton-Puiseux algorithm, which computes a feasible $\rho$ using $2^{O(m+n^2)}$ arithmetic operations.

math.AG↗

arXiv2022

Improved effective {\L}ojasiewicz inequality and applications

Let $\mathrm{R}$ be a real closed field. Given a closed and bounded semi-algebraic set $A \subset \mathrm{R}^n$ and semi-algebraic continuous functions $f,g:A \rightarrow \mathrm{R}$, such that $f^{-1}(0) \subset g^{-1}(0)$, there exist $N$ and $c \in \mathrm{R}$, such that the inequality ({\L}ojasiewicz inequality) $|g(x)|^N \le c \cdot |f(x)|$ holds for all $x \in A$. In this paper we consider the case when $A$ is defined by a quantifier-free formula with atoms of the form $P = 0, P >0, P \in \mathcal{P}$ for some finite subset of polynomials $\mathcal{P} \subset \mathrm{R}[X_1,\ldots,X_n]_{\leq d}$, and the graphs of $f,g$ are also defined by quantifier-free formulas with atoms of the form $Q = 0, Q >0, Q \in \mathcal{Q}$, for some finite set $\mathcal{Q} \subset \mathrm{R}[X_1,\ldots,X_n,Y]_{\leq d}$. We prove that the {\L}ojasiewicz exponent $N$ in this case is bounded by $(8 d)^{2(n+7)}$. Our bound depends on $d$ and $n$, but is independent of the combinatorial parameters, namely the cardinalities of $\mathcal{P}$ and $\mathcal{Q}$. As a consequence we improve the current best error bounds for polynomial systems under some conditions. Finally, as an abstraction of the notion of independence of the {\L}ojasiewicz exponent from the combinatorial parameters occurring in the descriptions of the given pair of functions, we prove a version of {\L}ojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common {\L}ojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions.

math.AG↗

arXiv2022

Sequents, barcodes, and homology

We consider the problem of generating hypothesis from data based on ideas from logic. We introduce a notion of barcodes, which we call sequent barcodes, that mirrors the barcodes in persistent homology theory in topological data analysis. We prove a theoretical result on the stability of these barcodes in analogy with similar results in persistent homology theory. Additionally we show that our new notion of barcodes can be interpreted in terms of a persistent homology of a particular filtration of topological spaces induced by the data. Finally, we discuss a concrete application of the sequent barcodes in a discovery problem arising from the area of cancer genomics.

math.AT↗

arXiv2022

Computing the homology functor on semi-algebraic maps and diagrams

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a semi-algebraic map $f:X \rightarrow Y$ between closed and bounded semi-algebraic sets. For every fixed $\ell \geq 0$ we give an algorithm with singly exponential complexity that computes bases of the homology groups $\mathrm{H}_i(X), \mathrm{H}_i(Y)$ (with rational coefficients) and a matrix with respect to these bases of the induced linear maps $\mathrm{H}_i(f):\mathrm{H}_i(X) \rightarrow \mathrm{H}_i(Y), 0 \leq i \leq \ell$. We generalize this algorithm to more general (zigzag) diagrams of maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.

math.AT↗

arXiv2022

Topology of real multi-affine hypersurfaces and a homological stability property

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to Petrovski{\u\i} and Ole{\u\i}nik, Thom and Milnor). Moreover, we show there exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a closed ball in $\mathrm{R}^n$ of positive radious, there exist hypersurfaces $(V_n)_{n_>0}$ defined by symmetric multi-affine polynomials of degree $4$, such that $\sum_{i \leq 5} b_i(V_n \cap B_n) > c^n$, where $b_i(\cdot)$ denotes the $i$-th Betti number with rational coeffcients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

math.AG↗

arXiv2022

Persistent homology of semi-algebraic sets

We give an algorithm with singly exponential complexity for computing the barcodes up to dimension $\ell$ (for any fixed $\ell \geq 0$) of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity, and generalizes the corresponding results for computing the Betti numbers up to dimension $\ell$ of semi-algebraic sets with no filtration present.

math.AT↗

arXiv2021

Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set

Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a field $\mathbb{F}$, of any given semi-algebraic set $S \subset \mathrm{R}^k$ defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset $\Gamma$ of the given semi-algebraic set $S$, such that $\mathrm{H}_q(S,\Gamma) = 0$ for $q=0,1$. We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety $X$ of dimension $n$, there exists Zariski closed subsets \[ Z^{(n-1)} \supset \cdots \supset Z^{(1)} \supset Z^{(0)} \] with $\dim_{\mathrm{C}} Z^{(i)} \leq i$, and $\mathrm{H}_q(X,Z^{(i)}) = 0$ for $0 \leq q \leq i$. We conjecture a quantitative version of this result in the semi-algebraic category, with $X$ and $Z^{(i)}$ replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of $Z^{(0)}$ and $Z^{(1)}$ with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing $Z_0$).

math.AG↗