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Mattie Ji

Publications and source records attributed to Mattie Ji.

14 recordsLinked to original sources
arXiv2026

$K$-Theoretic Obstructions to Linearizing QCA Representations

Projective representations arise naturally in physics and representation theory, and determining whether they can be linearized has been a fundamental problem. In this work, we study the analogous problem for quantum cellular automata (QCA) representations, which incorporate locality constraints imposed by a metric space $X$. Over an arbitrary field $\mathbb{F}$, we develop an obstruction theory for the linearization of QCA representations, using the algebraic $K$-theory spectrum of QCA constructed in previous work of the authors. The resulting obstructions are governed by the homotopy type of the QCA spaces, from which we extract universal obstruction classes to linearization. In the complex algebraic and unitary case, we also fully compute the homotopy types of the QCA spaces over a point, a line, and a plane.

math.AT↗
arXiv2026

Tensor Product $K$-theory is Rational Algebraic $K$-theory

For a commutative ring $R$ with unity, its algebraic $K$-theory space $K(R)$ may be obtained by group-completing the symmetric monoidal category of finitely generated free $R$-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of $K(R)$, up to $\pi_0$. We also discuss how a similar group-completion would give the $p$-perfection and, more generally, the localization of $K(R)$ at any non-trivial multiplicatively closed subset $S \subseteq \mathbb{Z}_{> 0}$. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.

math.AT↗
arXiv2026

Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.

cs.LG↗
arXiv2026

Quantum Cellular Automata: The Group, the Space, and the Spectrum

Over an arbitrary commutative ring $R$, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space $\mathbf{Q}(X)$ of quantum cellular automata (QCA) on a given metric space $X$. In most cases of interest, $\pi_0 \mathbf{Q}(X)$ classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences $\mathbf{Q}(*) \simeq \Omega^n \mathbf{Q}(\mathbb{Z}^n)$ for all $n$, which shows that the classification of QCA on Euclidean lattices is given by an $\Omega$-spectrum indexed by the dimension $n$. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya $R$-algebras, which may be of independent interest. We also include a section leading to the $\Omega$-spectrum for QCA over $C^*$-algebras with unitary circuits.

math.AT↗
arXiv2025

On topological descriptors for graph products

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex- and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations. Code is available at https://github.com/Aalto-QuML/tda_graph_product.

cs.LG↗
arXiv2025

Graph Persistence goes Spectral

Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than PH and spectral information on graphs alone. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.

cs.LG↗
arXiv2024

Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers

There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-Šmulian theorem). However, the existence theorems for many variational problems over non-reflexive Sobolev spaces remain underexplored. In this paper, we investigate various examples of functionals over non-reflexive Sobolev spaces. To do this, we prove a weak compactness criterion in $W^{k,1}$ that generalizes the Dunford-Pettis theorem, which asserts that relatively weakly compact subsets of $ L^1 $ coincide with equi-integrable families. As a corollary, we also extend an existence theorem of minimizers from reflexive Sobolev spaces to non-reflexive ones. This work is also benefited and streamlined by various concepts in category theory.

math.FA↗
arXiv2024

On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces

The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we first study the injectivity of the ECT on definable sets that are not necessarily compact and prove a complete classification of constructible functions that the Euler characteristic transform is not injective on. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the injectivity of QECT.

cs.CG↗
arXiv2023

Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis

Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right-continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+\delta}$ deformation retracts to $S^f_t$ for all sufficiently small $\delta > 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.

math.AT↗
arXiv2023

Explicit equations of the fake projective plane $(a=7,p=2,\emptyset,D_3 X_7)$

We find explicit equations of the fake projective plane $(a=7,p=2,\emptyset,D_3 X_7)$, which lies in the same class as the fake projective plane $(a=7,p=2,\emptyset,D_3 2_7)$ with $21$ automorphisms whose equations were previously found by Borisov and Keum. The method involves finding a birational model of a common Galois cover of these two surfaces.

math.AG↗
arXiv2023

Statistical Inference on Grayscale Images via the Euler-Radon Transform

Tools from topological data analysis have been widely used to represent binary images in many scientific applications. Methods that aim to represent grayscale images (i.e., where pixel intensities instead take on continuous values) have been relatively underdeveloped. In this paper, we introduce the Euler-Radon transform, which generalizes the Euler characteristic transform to grayscale images by using o-minimal structures and Euler integration over definable functions. Coupling the Karhunen-Loeve expansion with our proposed topological representation, we offer hypothesis-testing algorithms based on the chi-squared distribution for detecting significant differences between two groups of grayscale images. We illustrate our framework via extensive numerical experiments and simulations.

stat.ME↗
arXiv2023

On the Geometry of a Fake Projective Plane with $21$ Automorphisms

A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) = C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C} P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$. For each torsion line bundle $T \in \operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample generator of the N\'eron-Severi group.

math.AG↗
arXiv2023

Rationality of real conic bundles with quartic discriminant curve

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett--Tschinkel and Benoist--Wittenberg. The double cover models we consider were introduced and previously studied by S. Frei, S. Sankar, B. Viray, I. Vogt, and the first author.

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