We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $Δ_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $\varphi_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $Δ_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2024; arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (Combinatorica, 2016; arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obt
Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric topology, and combinatorial Laplacians have been proposed, combinatorial commutative algebra has hardly been developed for machine learning and data science. In this work, we introduce persistent Stanley-Reisner theory to bridge commutative algebra, combinatorial algebraic topology, machine learning, and data science. We propose persistent h-vectors, persistent f-vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules. Stability analysis indicates that these algebraic invariants are stable against geometric perturbations. We employ a machine learning prediction on a molecular dataset to demonstrate the utility of the proposed persistent Stanley-Reisner theory for practical applications.
Persistent tensors, introduced in [Quantum 8 (2024), 1238], and inspired by quantum information theory, form a recursively defined class of tensors that remain stable under the substitution method and thereby yield nontrivial lower bounds on tensor rank. In this work, we investigate the symmetric case-namely, symmetric persistent tensors, or equivalently, persistent polynomials. We establish that a symmetric tensor in $\mathrm{Sym}^n \mathbb{C}^d$ is persistent if the determinant of its Hessian equals the $d(n-2)$-th power of a nonzero linear form. The converse is verified for cubic tensors ($n=3$) or for $d \leq 3$, by leveraging classical results of B. Segre. Moreover, we demonstrate that the Hessian of a symmetric persistent tensor factors as the $d$-th power of a form of degree $(n-2)$. Our main results provide an explicit necessary and sufficient criterion for persistence, thereby offering an effective algebraic characterization of this class of tensors. Beyond characterization, we present normal forms in small dimensions, place persistent polynomials within prehomogeneous geometry, and connect them with semi-invariants, homaloidal polynomials, and Legendre transforms. Particu
Compute Express Link (CXL) enables memory pooling over disaggregated memory, offering the potential to improve resource utilization in persistent memory systems. However, integrating persistence semantics into CXL-based memory pooling introduces substantial latency, which limits system scalability. This overhead arises because persist operations must traverse the entire CXL fabric, including switches, links, and protocol layers, before reaching remote persistent memory. To this end, we argue that extending CXL switches with persistence support is a promising direction for improving the scalability of persistent memory pooling. However, moving persistence support into the network breaks the traditional correctness assumptions of centralized persistence domains. In particular, enabling persistence within distributed structures, such as CXL switches, can introduce stale reads and writes if not carefully coordinated. In this paper, we propose Distributed Persistence Domain (DPD), a new abstraction for persistent memory pooling that enables persistence support at the CXL switch level. We first formalize the concept of a distributed persistence domain and use DPD as a framework to identi
A $1$-Lipschitz map between compact metric spaces $f\colon X\to Y$ induces a homomorphism of persistence modules on degree-$d$ Vietoris--Rips persistent homology. We define the persistent cost of $f$ from this induced homomorphism by quantifying the persistence carried by its kernel and cokernel modules. We prove that the persistent cost controls the interleaving distance between the degree-$d$ Vietoris--Rips persistent homology modules of $X$ and $Y$. Moreover, we obtain an explicit upper bound for the persistent cost in purely metric terms. Finally, we give a self-contained proof of the stability of the persistent cost introducing a Gromov-Hausdorff type distance for maps between compact metric spaces.
Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the ability to reliably detect more complex topologies, particularly toroidal structures, despite previous heuristic attempts. To address this limitation, recent work introduced persistent cup-length, a novel topological invariant derived from persistent cohomology. In this paper, we present the first implementation of this method and demonstrate its practical effectiveness in detecting toroidal structures, uncovering topological features beyond the reach of persistent homology alone. Our implementation overcomes computational bottlenecks through strategic optimization, efficient integration with the Ripser library, and the application of landmark subsampling techniques. By applying our method to grid cell population activity data, we demonstrate that persistent cup-length effectively identifies toroidal structures in neural manifolds. Our approach offers a powerful new tool for analyzing high-dimensional data, advancing existing topological methods.
Background: Large language models are typically evaluated as models, benchmarks, or short conversational episodes. Less is known about what happens when an agent is embedded persistently in a real academic research environment with durable memory, local files, external tools, scheduled routines, delegated roles, and explicit safety protocols. Methods: A structured self-observed implementation case study was conducted from January 31 to May 25, 2026. The unit of analysis was the persistent human-agent environment: researcher, agent runtime, memory layer, tools, repositories, scheduled jobs, specialized agent roles, and governance rules. Outcomes were organized using PARE-M (Persistent Agentic Research Environment Measurement), a measurement framework covering architecture, utilization, artifact production, resource use, reproducibility, and governance. Results: Recoverable main-agent telemetry contained 75,671 de-duplicated records across 96 active days, with 8,059 user-role and 23,710 assistant-role messages. The workspace included 502 memory-related files, 17 configured agent directories, and 57 skill files. Active system time was 579.7 hours (30-minute capped-gap estimate). Memor
Compute Express Link (CXL) switch allows memory extension via PCIe physical layer to address increasing demand for larger memory capacities in data centers. However, CXL attached memory introduces 170ns to 400ns memory latency. This becomes a significant performance bottleneck for applications that host data in persistent memory as all updates, after traversing the CXL switch, must reach persistent domain to ensure crash consistent updates. We make a case for persistent CXL switch to persist updates as soon as they reach the switch and hence significantly reduce latency of persisting data. To enable this, we presented a system independent persistent buffer (PB) design that ensures data persistency at CXL switch. Our PB design provides 12\% speedup, on average, over volatile CXL switch. Our \textit{read forwarding} optimization improves speedup to 15\%.
Feature extraction in noisy image datasets presents many challenges in model reliability. In this paper, we use the discrete Fourier transform in conjunction with persistent homology analysis to extract specific frequencies that correspond with certain topological features of an image. This method allows the image to be compressed and reformed while ensuring that meaningful data can be differentiated. Our experimental results show a level of compression comparable to that of using JPEG using six different metrics. The end goal of persistent homology-guided frequency filtration is its potential to improve performance in binary classification tasks (when augmenting a Convolutional Neural Network) compared to traditional feature extraction and compression methods. These findings highlight a useful end result: enhancing the reliability of image compression under noisy conditions.
Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combines multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in analyzing large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, as well as $N$-chain complexes.
Persistent language-model agents increasingly combine tool use, tiered memory, reflective prompting, and runtime adaptation. In such systems, behavior is shaped not only by current prompts but by mutable internal conditions that influence future action. This paper introduces layered mutability, a framework for reasoning about that process across five layers: pretraining, post-training alignment, self-narrative, memory, and weight-level adaptation. The central claim is that governance difficulty rises when mutation is rapid, downstream coupling is strong, reversibility is weak, and observability is low, creating a systematic mismatch between the layers that most affect behavior and the layers humans can most easily inspect. I formalize this intuition with simple drift, governance-load, and hysteresis quantities, connect the framework to recent work on temporal identity in language-model agents, and report a preliminary ratchet experiment in which reverting an agent's visible self-description after memory accumulation fails to restore baseline behavior. In that experiment, the estimated identity hysteresis ratio is 0.68. The main implication is that the salient failure mode for persi
The appearance of highly anisotropic planes of satellites around the Milky Way and other galaxies was long considered a challenge to the standard cosmological model. Some recent simulations have found flattened satellite systems to be common, but these have been described as either "transient", short-lived alignments, or "persistent", long-lived structures. Here we analyse Milky Way analogue systems in the cosmological simulation TNG-50 to resolve this apparent contradiction. We show that, as the satellite populations of individual hosts rapidly change, the observed spatial anisotropies of their satellite systems are invariably short-lived, with lifetimes of no more than a few hundred million years. However, when the progenitors of the same satellites are traced backwards, we find examples where those identified to form a plane at the present day have retained spatial coherence over several billion years. The two ostensibly conflicting predictions for the lifetimes of satellite planes can be reconciled as two perspectives on the same phenomenon.
We study the foundational properties of persistent homotopy groups and develop elementary computational methods for their analysis. Our main theorems are persistent analogues of the Van Kampen, excision, suspension, and Hurewicz theorems. We prove a persistent excision theorem, derive from it a persistent Freudenthal suspension theorem, and obtain a persistent Hurewicz theorem relating the first nonzero persistent homotopy group of a space to its persistent homology. As an application, we compute sublevelset persistent homotopy groups of alkane energy landscapes and show these invariants capture nontrivial loops and higher-dimensional features that comple- ment the information given by persistent homology.
The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of persistent Laplacians has emerged as a novel approach to topological persistence, attracting significant attention and finding applications in various fields. In this paper, we investigate the stability of persistent Laplacians. We introduce the notion of ``Laplacian trees'', which captures the collection of persistent Laplacians that persist from a given parameter. To formalize our study, we construct the category of Laplacian trees and establish an algebraic stability theorem for persistent Laplacian trees. Notably, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such topological features is often a rather formidable task necessitating the subsampling the underlying data. To remedy this, we develop an efficient quantum computation of persistent Betti numbers, which track topological features of data across different scales. Our approach employs a persistent Dirac operator whose square yields the persistent combinatorial Laplacian, and in turn the underlying persistent Betti numbers which capture the persistent features of data. We also test our algorithm on point cloud data.
Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in diverse domains. Despite its widespread use, persistent homology is simply impossible to compute when a dataset is very large. We study a statistical approach to the problem of computing persistent homology for massive datasets using a multiple subsampling framework and extend it to three summaries of persistent homology: Hölder continuous vectorizations of persistence diagrams; the alternative representation as persistence measures; and standard persistence diagrams. Specifically, we derive finite sample convergence rates for empirical means for persistent homology and practical guidance on interpreting and tuning parameters. We validate our approach through extensive experiments on both synthetic and real-world data. We demonstrate the performance of multiple subsampling in a permutation test to analyze the topological structure of Poincaré embeddings of large lexical databases.
We present a thorough study of the theoretical properties and devise efficient algorithms for the \emph{persistent Laplacian}, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes $K \hookrightarrow L$, which was independently introduced by Lieutier et al. and by Wang et al. In particular, in analogy with the non-persistent case, we first prove that the nullity of the $q$-th persistent Laplacian $Δ_q^{K,L}$ equals the $q$-th persistent Betti number of the inclusion $(K \hookrightarrow L)$. We then present an initial algorithm for finding a matrix representation of $Δ_q^{K,L}$, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional, very simple algorithm for finding (a matrix representation of) the $q$-th persistent Laplacian which in turn leads to a novel and fund
Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper we formalize this idea, and prove the stability of the persistent Betti number functions in G-invariant persistent homology with respect to the natural pseudo-distance d_G. We also show how G-invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self-homeomorphisms of a topological space. In this paper we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without u
Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the stability theorem with respect to real-valued functions on a topological space is one of the fundamental results, which gives a mathematical justification for using persistent homology to noisy data. In proving the stability theorem, the algebraic stability theorem/the isometry theorem for persistence modules plays a central role. In this point of view, the stability property for bipath persistent homology is desired for analyzing data. In this paper, we prove the stability theorem of bipath persistent homology with respect to bipath functions on a topological space. This theorem suggests a stability of bipath persistence diagrams: small changes in a bipath function (except at their ends) result in only small changes in the bipath persistence diagram. Similar to the stability theorem of standard persistent homology, we deduce the stability theorem of bipath persistent homology by using the algebraic stability theorem/the isometry theorem of bipa
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov-Hausdorff distance. We also present examples illustrating that persistent topological complexity and persistent zero-divisor-cup-length can distinguish between certain spaces more effectively than persistent homology.