Proposes CXNs for neural network computations on cell complexes.
problem Performing neural network computations on complex topological spaces.
method Introduces a message passing scheme and a unified encoder-decoder framework for cell complexes.
result Generalizes message passing to cell complexes and provides a cell2vec representation.
Minimal example found for two finite CW-complexes sharing a common covering.
problem Finding the minimum number of 2-cells in two finite CW-complexes that share a common covering.
method Constructing an almost minimal example with two 2-cells in each complex.
result Minimal example with two 2-cells in each complex.
Paper introduces signal processing on cell complexes.
problem Processing signals on non-Euclidean domains.
method Signal processing on abstract regular cell complexes.
result Hodge Laplacians for cell complexes enable convolutional filters.
Study homology of periodic cell complexes using quotient spaces and spectral sequences.
problem Quantifying homology in periodic cell complexes.
method Finite representation of periodic cell complexes, Mayer-Vietoris spectral sequence.
result Full recovery of homology generators for d-periodic graphs. We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then …
CT improves neural network performance on cell complex data.
problem Improving predictive performance of neural networks on complex data.
method Introducing the Cellular Transformer (CT) that generalizes graph-based transformers to cell complexes.
result CT achieves state-of-the-art performance on cell complex datasets without complex enhancements.
In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochne…
The paper introduces optimal transport kernels for comparing cell complexes.
problem Lack of machine learning methods for CW complexes.
method Derives explicit expression for Wasserstein distance, extends Fused Gromov-Wasserstein, introduces novel kernels.
result Introduced novel kernels for comparing probability measures on CW complexes.
Establishing criteria for top cell inertness in complexes.
problem Criteria for top cell inertness in Poincaré duality complexes.
method Algebraic intersection theory, homotopy fibrations, surgery, homogeneous spaces.
result Established various criteria for top cell inertness.
Defines a cell complex for even spin mapping class group.
problem No specific problem stated; focuses on definition.
method Defines a cell complex with an action of the even spin mapping class group.
result Obtains a finite presentation of the even spin mapping class group.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
The study identifies all possible vector field structures on specific 2D shapes.
problem Optimal discrete gradient vector fields on surfaces with 1-2 critical cells.
method Analysis of discrete vector fields on 2D shapes with minimal critical cells.
result All possible structures of discrete Morse functions on specified shapes.
Two simple homotopy equivalent 2-complexes K2 and L2 are related by an algebraic criterion of their corresponding presentations as stated in [HoMeSier]. Frank Quinn set it into a topological context (see [Qu1]) and call these 2-complexes related by an s-move. Using elementary 3-expansions, K2 extends to 3-cells in K3 r…
A new method for state estimation on complex networks.
problem Reconstructing latent dynamics from multivariate time-series on topological cell complexes.
method Topology-aware state space framework derived from stochastic partial differential equations, with state evolution following heat-like topological diffusion.
result The proposed method successfully recovers latent states and topological structures in real-world networks.
Advances combinatorial complexes for better modeling of hierarchical and set-type relations.
problem Lack of effective modeling for complex hierarchical and set-type relations in high-dimensional data.
method Introduces combinatorial complexes as a bridge between cell complexes and hypergraphs, emphasizing their different types of relations.
result Combining set-type and hierarchical relations in a single model can be advantageous in learning tasks.
The study describes a cell structure for multisets in a rectangle.
problem Understanding the space of multisets in a rectangle.
method Developed a piecewise Euclidean bi-simplicial cell structure.
result Connected to spaces of complex polynomials and permutahedra.
A new model for complex cells accounts for insensitivity to image shifts.
problem Complex cells' response to image shifts.
method Linear combination of Gaussian derivatives at a single position.
result Maximum response insensitive to small shifts of the image.
Two complexes share a common covering but not a finite one.
problem Common covering of complexes with specific properties.
method Constructing CW-complexes K and L with a common covering.
result K is homeomorphic to a complex with a single 2-cell.
Constructs a simplicial cell decomposition of complex projective space for n ≥ 2.
problem Finding a simplicial cell decomposition for complex projective space.
method Starting with a standard crystallisation of the 2-sphere, constructing a simplicial subdivision, and quotienting by the Sym(n) action.
result Explicit construction of a simplicial cell decomposition of complex projective space for n ≥ 2.
Short proofs for complex Tverberg theorems using prime powers.
problem Proving Tverberg-type theorems for cell complexes.
method Short proofs using prime powers and continuous maps.
result Pairwise disjoint faces of a complex intersect under continuous maps.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
New rational curvature measures for 2-complexes.
problem Measuring curvature in 2-dimensional cell complexes.
method Defined and proved rational curvature invariants.
result Computable rational curvature bounds for 2-complexes.
Proposes CCCVAE for better single-cell clustering with cell-cell communication.
problem Improving single-cell RNA sequencing clustering by incorporating cell-cell communication.
method Integrates cell-cell communication into a variational autoencoder framework.
result Empirical results show CCCVAE outperforms standard VAEs in clustering performance.
The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposit…
Chaos in cerebellar cells enhances complexity of neural patterns.
problem Understanding how cerebellar granular layer represents complex information.
method Constructed a model of cerebellar granular layer with gap junctions, evaluated using reservoir computing.
result Chaotic dynamics in the cerebellar granular layer produce complex and diverse output patterns.
This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally non…
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …
The paper studies geometric structures of polynomial spaces.
problem Understanding the geometric and combinatorial structures of polynomial spaces.
method Introducing and analyzing finite piecewise Euclidean cell complexes.
result The branched rectangle and annulus complexes are homeomorphic to specific polynomial spaces.
Cell-based NAS search spaces are redundant and lack novelty.
problem Redundancy and lack of novelty in cell-based neural architecture search.
method Empirical analysis of architectures from popular cell-based search spaces.
result Randomly sampled architectures can match or outperform state-of-the-art results.
We study Lagrangian embeddings of a class of two-dimensional cell complexes Lp,q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type p21(pq−1,1) (Wahl singularities). We show that …
Complex numbers have long been favoured for digital signal processing, yet complex representations rarely appear in deep learning architectures. RNNs, widely used to process time series and sequence information, could greatly benefit from complex representations. We present a novel complex gated recurrent cell, which i…
We develop a latent variable model and an efficient spectral algorithm motivated by the recent emergence of very large data sets of chromatin marks from multiple human cell types. A natural model for chromatin data in one cell type is a Hidden Markov Model (HMM); we model the relationship between multiple cell types by…
The paper extends Gaussian processes to model complex interactions in cellular complexes.
problem Capturing topological inductive biases in machine learning models.
method Proposes Gaussian processes on cellular complexes, introducing novel kernels.
result Derives two novel kernels for modeling interactions between cells.
This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.
Tree complex linked to polyhedral shapes like associahedra and cyclohedra.
problem Understanding the structure of mapping class groups and complex dynamics.
method Characterizing associahedra and cyclohedra using planar tree embeddings and barycentric subdivision.
result Tree complex is a barycentric subdivision of a polyhedral cell complex made of associahedra and cyclohedra.
We posit that hippocampal place cells encode information about future locations under a transition distribution observed as an agent explores a given (physical or conceptual) space. The encoding of information about the current location, usually associated with place cells, then emerges as a necessary step to achieve t…
New method learns complex cell networks from millions of cells.
problem Existing methods fail to scale to large datasets.
method Multi-axis Gaussian graphical models.
result Method scales to millions of cells in minutes.
Develops a new neural spike train decoding framework using topological data.
problem Decoding neural spike trains from head direction and grid cells.
method Combines simplicial complex discovery with deep learning to capture higher-order connectivity.
result Demonstrates effectiveness on head direction and trajectory prediction datasets.
Compact Kähler manifold minus a divisor is projective space.
problem Compact Kähler manifolds minus smooth divisors.
method Analyzing homology cells and applying Fujita's conjectures.
result Compact Kähler manifolds minus contractible divisors are projective spaces.
Understanding cell identity is an important task in many biomedical areas. Expression patterns of specific marker genes have been used to characterize some limited cell types, but exclusive markers are not available for many cell types. A second approach is to use machine learning to discriminate cell types based on th…
Neuronal circuits formed in the brain are complex with intricate connection patterns. Such complexity is also observed in the retina as a relatively simple neuronal circuit. A retinal ganglion cell receives excitatory inputs from neurons in previous layers as driving forces to fire spikes. Analytical methods are requir…
Multi-StyleGAN simulates live cell microscopy imagery.
problem Costly and complex live cell experiments.
method Generative adversarial network (GAN) synthesizing multi-domain time-lapse images.
result Captures biophysical factors and time dependencies in cell imagery.
Novel algorithm learns sparse signal representations over topological spaces.
problem Sparse representation of signals over combinatorial topological spaces.
method Leveraging Hodge theory, the paper embeds topology into a dictionary structure via concatenated sub-dictionaries, each as a polynomial of Hodge Laplacians, and optimizes the dictionary coefficients and sparse signal representation via iterative alternating algorithms.
result Efficiently learned sparse representations and underlying relational structure of topological signals.
GENOT matches cells across data modalities using neural OT solvers.
problem Scalability, privacy, and out-of-sample estimation issues in traditional OT solvers.
method Learn stochastic maps, parameterize OT maps, relax mass conservation, integrate quadratic solvers.
result Demonstrates significant potential for enhancing therapeutic strategies.
Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this me…
Proposes a model for identifying 4G cells with network throughput problems.
problem Challenges in identifying 4G cells with network throughput issues due to network complexity and privacy concerns.
method Data-driven model using clustering and Deep Neural Networks (DNNs). Model parameters are learned from a small number of expert-labeled data. Multiple clustering models capture common features for problematic cells.
result The proposed model outperforms a simple classifier in identifying cells with network throughput problems.
New model generates realistic single-cell gene expression data.
problem Generating realistic single-cell gene expression profiles is challenging.
method scLDM, a latent diffusion model using Diffusion Transformers and linear interpolants.
result Superior performance in generating realistic single-cell gene expression data.
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.