Higher Dimensional Manifolds Research Articles

Mind the gap: hole-filling and reconstruction of high-dimensional manifolds from noisy scattered data

Mind the gap: hole-filling and reconstruction of high-dimensional manifolds from noisy scattered data

Incremental Classification for High-Dimensional EEG Manifold Representation Using Bidirectional Dimensionality Reduction and Prototype Learning.

In brain-computer interface (BCI) systems, symmetric positive definite (SPD) manifold within Riemannian space has been frequently utilized to extract spatial features from electroencephalogram (EEG) signals. However, the intrinsic high dimensionality of SPD matrices introduces too much computational burden to hinder the real-time applications of such BCI, especially in handling dynamic tasks, like incremental learning. Directly reducing the dimensionality of SPD matrices with conventional dimensionality reduction (DR) methods will alter the fundamental properties of SPD matrices. Moreover, current DR methods for incremental learning always necessitate retaining old data to update their representations under new mapping. To this end, a bidirectional two-dimensional principal component analysis for SPD manifold (B2DPCA-SPD) is proposed to reduce the dimensionality of SPD matrices, in such way that the reduced matrices remain on SPD manifold. Afterwards, the B2DPCA-SPD is extended to adapt to incremental learning tasks without saving old data. The incremental B2DPCA-SPD can be seamlessly integrated with the matrix-formed growing neural gas network (MF-GNG) to achieve an incremental EEG classification, where the new low-dimensional representations of the prototypes in old classifiers can be easily recalculated with the updated projection matrix. Extensive experiments are conducted on two public datasets to perform the EEG classification. The results demonstrate that our method significantly reduces computation time by 38.53% and 35.96%, and outperforms conventional methods in classification accuracy by 4.21% to 19.59%.

Algebraic and Giroux torsion in higher-dimensional contact manifolds

Algebraic and Giroux torsion in higher-dimensional contact manifolds

High-dimensional latent manifolds as predictors of individual differences in naturalistic movie viewing

High-dimensional latent manifolds as predictors of individual differences in naturalistic movie viewing

Mesoscopic structure graphs for interpreting uncertainty in non-linear embeddings

Probabilistic-based non-linear dimensionality reduction (PB-NL-DR) methods, such as t-SNE and UMAP, are effective in unfolding complex high-dimensional manifolds, allowing users to explore and understand the structural patterns of data. However, due to the trade-off between global and local structure preservation and the randomness during computation, these methods may introduce false neighborhood relationships, known as distortion errors and misleading visualizations. To address this issue, we first conduct a detailed survey to illustrate the design space of prior layout enrichment visualizations for interpreting DR results, and then propose a node-link visualization technique, ManiGraph. This technique rethinks the neighborhood fidelity between the high- and low-dimensional spaces by constructing dynamic mesoscopic structure graphs and measuring region-adapted trustworthiness. ManiGraph also addresses the overplotting issue in scatterplot visualization for large-scale datasets and supports examining in unsupervised scenarios. We demonstrate the effectiveness of ManiGraph in different analytical cases, including generic machine learning using 3D toy data illustrations and fashion-MNIST, a computational biology study using a single-cell RNA sequencing dataset, and a deep learning-enabled colorectal cancer study with histopathology-MNIST.

Symbolic Dynamics for Nonuniformly Hyperbolic Maps with Singularities in High Dimension

We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.

MDFL: Multi-Domain Diffusion-Driven Feature Learning

High-dimensional images, known for their rich semantic information, are widely applied in remote sensing and other fields. The spatial information in these images reflects the object's texture features, while the spectral information reveals the potential spectral representations across different bands. Currently, the understanding of high-dimensional images remains limited to a single-domain perspective with performance degradation. Motivated by the masking texture effect observed in the human visual system, we present a multi-domain diffusion-driven feature learning network (MDFL) , a scheme to redefine the effective information domain that the model really focuses on. This method employs diffusion-based posterior sampling to explicitly consider joint information interactions between the high-dimensional manifold structures in the spectral, spatial, and frequency domains, thereby eliminating the influence of masking texture effects in visual models. Additionally, we introduce a feature reuse mechanism to gather deep and raw features of high-dimensional data. We demonstrate that MDFL significantly improves the feature extraction performance of high-dimensional data, thereby providing a powerful aid for revealing the intrinsic patterns and structures of such data. The experimental results on three multi-modal remote sensing datasets show that MDFL reaches an average overall accuracy of 98.25%, outperforming various state-of-the-art baseline schemes. Code available at https://github.com/LDXDU/MDFL-AAAI-24.

Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism.

Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

Existence of harmonic maps and eigenvalue optimization in higher dimensions

AbstractWe prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^{n},g)$ ( M n , g ) of dimension $n>2$ n > 2 to any closed, non-aspherical manifold $N$ N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^{k}$ N = S k , $k\geqslant 3$ k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^{k}$ M → S k of index at most $k+1$ k + 1 , with singular set of codimension at least 7 for $k$ k sufficiently large. Furthermore, if $3\leqslant n\leqslant 5$ 3 ⩽ n ⩽ 5 , we show that these smooth harmonic maps stabilize as $k$ k becomes large, and correspond to the solutions of an eigenvalue optimization problem on $M$ M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

Automorphic spectra and the conformal bootstrap

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of P S L 2 ( R ) \mathrm {PSL}_2(\mathbb {R}) and semidefinite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is λ 1 ≤ 3.8388976481 \lambda _1\leq 3.8388976481 , while the Bolza surface has λ 1 ≈ 3.838887258 \lambda _1\approx 3.838887258 . The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

Minkowski Functionals in 𝖲𝖮(3) for the spin-2 CMB polarisation field

The study of the angular power spectrum of Cosmic Microwave Background (CMB) anisotropies, both in intensity and in polarisation, has led to the tightest constraints on cosmological parameters. However, this statistical quantity is not sensitive to any deviation from Gaussianity and statistical isotropy in the CMB data. Minkowski Functionals (MFs) have been adopted as one of the most powerful statistical tools to study such deviations, since they characterise the topology and geometry of the field of interest. In this paper, we extend the application of MFs to CMB polarisation data by introducing a new formalism, where we lift the spin 2 polarisation field to a scalar function in a higher-dimensional manifold: the group of rotations of the sphere, SO(3). Such a function is defined as f = Q cos(2ζ) - U sin(2ζ). We analytically obtain the expected values for the MFs of f in the case of Gaussian isotropic polarisation maps. Furthermore, we present a new pipeline which estimates these MFs from input HEALPix polarisation maps. We apply it to CMB simulations in order to validate the theoretical results and the methodology. The pipeline is to be included in the publicly available Python package Pynkowski .

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

We present the Julia package Manifolds.jl , providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase.jl , with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds.

Maps from 3-manifolds to 4-manifolds that induce isomorphisms on π1

In this paper, we prove that if a closed orientable [Formula: see text]-manifold [Formula: see text] admits a map [Formula: see text] to a closed orientable [Formula: see text]-manifold [Formula: see text] such that [Formula: see text] induces an isomorphism on fundamental groups, then [Formula: see text] is homeomorphic to [Formula: see text] or [Formula: see text]. Relevant results on higher dimensional manifolds are also obtained.

EPCO-12. IN VIVO PERTURB-SEQ DEFINES BIOLOGICAL DRIVERS AND THERAPEUTIC VULNERABILITIES UNDERLYING RADIOTHERAPY RESISTANCE IN GLIOBLASTOMA

Abstract Glioblastomas are comprised of dynamic malignant cell states and microenvironment cell types. Mechanistic and functional studies of glioblastoma resistance to therapy have been limited by a lack of in vivo approaches for multiplexed interrogation of single cells. Here we use genome-wide and targeted CRISPR interference (CRISPRi) to identify therapeutic vulnerabilities in glioblastomas in vivo by coupling convection enhanced delivery (CED) with functional genomics and single-cell transcriptomics (perturb-seq). Human (GBM43, LN18) or mouse (GL261, SB28) glioblastomas expressing CRISPRi machinery were established intracranially in mice, and unmodified mouse glioblastomas were established intracranially in immunocompetent mice expressing CRISPRi machinery in the tumor microenvironment. Perturb-seq gene prioritization was performed using genome-wide CRISPRi screens ± radiotherapy in cell cultures. Dual-sgRNA Perturb-seq lentivirus libraries were transduced in vivo using CED, and single-cell RNA sequencing with sgRNA capture was performed ± radiotherapy. RESULTS: were validated using RNA sequencing of 254,288 single nuclei from 43 pairs of primary and recurrent patient-matched human glioblastomas. Perturb-seq ± radiotherapy of 98 genes underlying glioblastoma radiotherapy responses was performed across 414,092 single cells in vivo. Genetic perturbations reprogrammed tumor and microenvironment cell states in response to radiotherapy, and high dimensional manifolds revealed genetic dependencies in metabolic and DNA damage response pathways. The DNA-dependent protein kinase (DNA-PK) component PRKDC was identified as a driver of glioblastoma radiotherapy resistance through regulation of malignant cell-intrinsic growth and stress pathways, and regulation of malignant cell-extrinsic interferon pathways that controlled cell-cell interactions with the tumor microenvironment. PRKDC inhibition with nedisertib sensitized glioblastomas to radiotherapy and extended survival in vivo, and single-cell RNA sequencing revealed hyperactivation of cell stress and cytokine signatures with combination treatment. In summary, we report in vivo perturb-seq as a platform for simultaneous discovery and functional interrogation of therapeutic vulnerabilities in glioblastoma, and show DNA-PK inhibition sensitizes glioblastomas to radiotherapy in vivo.

Identifying Gene-Treatment Interactions and Targetable Radiation Vulnerabilities in Glioblastoma through Coupling of In Vivo CRISPR Perturbation and Single Cell Transcriptomics

Glioblastoma (GBM) is an incurable brain tumor comprised of dynamic malignant cell states and microenvironment components that underlie treatment resistance. Here we use genome-wide CRISPR/Cas9 functional genomics to define biological drivers and therapeutic vulnerabilities across human and mouse GBM models. To interrogate these mechanisms in the context of the tumor microenvironment and in vivo physiology, we established in vivo Perturb-seq intracranially, a technique coupling functional genomics with single cell transcriptomics, where each cell is an individual experiment. Orthotopic intracranial tumor models were established using human (GBM6, GBM43) or mouse (GL261, SB28) GBM cells stably expressing CRISPR interference (CRISPRi) machinery. Perturb-seq target selection for phenotyping of gene-treatment interactions was performed using genome-wide CRISPRi screens ± radiotherapy in cell cultures. Dual sgRNA lentivirus libraries were transduced either ex vivo prior to intracranial GBM cell transplantation or in vivo using intratumor convection enhanced delivery (CED). Transcriptional phenotyping was performed using single-cell RNA-seq with CRISPR direct capture following focal brain radiotherapy (2 Gy x 5) or mock treatment. GBM cell states were validated using single-nucleus RNA-seq data from 86 primary-recurrent patient-matched GBMs. Mechanistic and functional validation was performed using small molecule inhibitors, immunohistochemistry, clonogenic assays, and in vivo survival experiments. In vivo Perturb-seq ± radiotherapy of 48 genes underlying GBM radiotherapy responses, which were enriched for DNA damage response and metabolic pathways, was performed in > 425,000 single cells. Radiotherapy induced 16 distinct GBM cell states, and genetic perturbations reprogrammed these cell states in a treatment-dependent fashion. Quantitative modeling of gene/radiotherapy interactions using high dimensional manifolds revealed in vivo-specific genetic dependencies. We revealed a critical role for Prkdc, the catalytic subunit of DNA-dependent protein kinase (DNA-PK), as a radiotherapy sensitizer through regulation of cell intrinsic growth and oxidative stress pathways, and cell extrinsic interferon and signaling pathways that altered cell-cell interactions in vivo. These pathways were also disrupted in single-nucleus RNA-seq analysis of post-radiotherapy human GBM tumors. Inhibition of Prkdc using a Food and Drug Administration approved small molecule sensitized GBM cells to radiotherapy and extended survival in mice harboring human intracranial xenografts. We establish in vivo Perturb-seq in orthotopic GBM models as a platform for simultaneous functional genomic discovery and characterization of therapeutic targets, revealing an underappreciated role for Prkdc in GBM tumors in vivo that is targetable using small molecules. These tools are adaptable for a wide range of disease models and treatment modalities.

Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along J-symplectic submanifolds. We also study branched coverings in the generalized complex setting. As an application, we produce stable generalized complex structures on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.

Application of high-dimensional uniform manifold approximation and projection (UMAP) to cluster existing landfills on the basis of geographical and environmental features

Due to extreme conditions, which are influenced by the location of landfills, the release of pollutants has been recently proven to be more severe in estuary landfills, as these landfill locations are affected by both sea-water and river-water interactions. To identify geographic and environmental features linked to the extreme conditions of certain landfills, a high-dimensional clustering method combining Uniform Manifold Approximation and Projection (UMAP) with the Louvain algorithm is proposed.A case study was conducted using 17 noteworthy features that transform to Landfill Suitability Index (LSI) applied to hundreds of landfill sites in Taiwan. This study clustered landfills into 10 clusters and identified several clusters with significant extreme locations, including estuary landfills (7.9 %), fault-water-body landfills (8.2 %), and densely-populated-water-body landfills (17.6 %). Furthermore, a critical discovery of endangered Platalea minor habitats near these estuary landfills was made. Additionally, this work identified “healthy” landfills (11.2 %) that are minimally affected by the considered features. These findings demonstrate the promising potential of our framework for managers to systematically improve landfill management strategies. Moreover, our framework was tested by incorporating rainfall and flooding features in relation to climate change scenarios. To address the demand for land release from occupied landfills in Taiwan, there is a pressing need to expedite the transition to a circular economy, and our framework can provide further assistance in this regard. This approach is promising, as it provides a new method to evaluate the environmental risks linked to landfills and also identifies potential opportunities related to landfill mining.Finally, this work was extended to include a case study in England, which has 19,801 landfills and a dataset containing 15 relevant landfill features; in this case study, our framework identified 110 landfill clusters, and several placed in extreme locations, demonstrating that our framework is flexible for use in other regions outside of Taiwan.

Penalized model-based clustering of complex functional data

High dimensional data, large-scale data, imaging and manifold data are all fostering new frontiers of statistics. These type of data are commonly considered in Functional Data Analysis where they are viewed as infinite-dimensional random vectors in a functional space. The rapid development of new technologies has generated a flow of complex data that have led to the development of new modeling strategies by scientists. In this paper, we basically deal with the problem of clustering a set of complex functional data into homogeneous groups. Working in a mixture model-based framework, we develop a flexible clustering technique achieving dimensionality reduction schemes through an L_1 penalization. The proposed procedure results in an integrated modelling approach where shrinkage techniques are applied to enable sparse solutions in both the means and the covariance matrices of the mixture components, while preserving the underlying clustering structure. This leads to an entirely data-driven methodology suitable for simultaneous dimensionality reduction and clustering. The proposed methodology is evaluated through a Monte Carlo simulation study and an empirical analysis of real-world datasets showing different degrees of complexity.

Stratification of diabetes in the context of comorbidities, using representation learning and topological data analysis

Diabetes is a heterogenous, multimorbid disorder with a large variation in manifestations, trajectories, and outcomes. The aim of this study is to validate a novel machine learning method for the phenotyping of diabetes in the context of comorbidities. Data from 9967 multimorbid patients with a new diagnosis of diabetes were extracted from Clinical Practice Research Datalink. First, using BEHRT (a transformer-based deep learning architecture), the embeddings corresponding to diabetes were learned. Next, topological data analysis (TDA) was carried out to test how different areas in high-dimensional manifold correspond to different risk profiles. The following endpoints were considered when profiling risk trajectories: major adverse cardiovascular events (MACE), coronary artery disease (CAD), stroke (CVA), heart failure (HF), renal failure (RF), diabetic neuropathy, peripheral arterial disease, reduced visual acuity and all-cause mortality. Kaplan Meier curves were plotted for each derived phenotype. Finally, we tested the performance of an established risk prediction model (QRISK) by adding TDA-derived features. We identified four subgroups of patients with diabetes and divergent comorbidity patterns differing in their risk of future cardiovascular, renal, and other microvascular outcomes. Phenotype 1 (young with chronic inflammatory conditions) and phenotype 2 (young with CAD) included relatively younger patients with diabetes compared to phenotypes 3 (older with hypertension and renal disease) and 4 (older with previous CVA), and those subgroups had a higher frequency of pre-existing cardio-renal diseases. Within ten years of follow-up, 2592 patients (26%) experienced MACE, 2515 patients (25%) died, and 2020 patients (20%) suffered RF. QRISK3 model’s AUC was augmented from 67.26% (CI 67.25–67.28%) to 67.67% (CI 67.66–67.69%) by adding specific TDA-derived phenotype and the distances to both extremities of the TDA graph improving its performance in the prediction of CV outcomes. We confirmed the importance of accounting for multimorbidity when risk stratifying heterogenous cohort of patients with new diagnosis of diabetes. Our unsupervised machine learning method improved the prediction of clinical outcomes.

ARKGraph: All-Range Approximate K-Nearest-Neighbor Graph

Given a collection of vectors, the approximate K-nearest-neighbor graph (KGraph for short) connects every vector to its approximate K-nearest-neighbors (KNN for short). KGraph plays an important role in high dimensional data visualization, semantic search, manifold learning, and machine learning. The vectors are typically vector representations of real-world objects (e.g., images and documents), which often come with a few structured attributes, such as times-tamps and locations. In this paper, we study the all-range approximate K-nearest-neighbor graph (ARKGraph) problem. Specifically, given a collection of vectors, each associated with a numerical search key (e.g., a timestamp), we aim to build an index that takes a search key range as the query and returns the KGraph of vectors whose search keys are within the query range. ARKGraph can facilitate interactive high dimensional data visualization, data mining, etc. A key challenge of this problem is the huge index size. This is because, given n vectors, a brute-force index stores a KGraph for every search key range, which results in O (K n 3 ) index size as there are O ( n 2 ) search key ranges and each KGraph takes O (K n ) space. We observe that the KNN of a vector in nearby ranges are often the same, which can be grouped together to save space. Based on this observation, we propose a series of novel techniques that reduce the index size significantly to just O (K n log n ) in the average case. Furthermore, we develop an efficient indexing algorithm that constructs the optimized ARKGraph index directly without exhaustively calculating the distance between every pair of vectors. To process a query, for each vector in the query range, we only need O (log log n + K log K) to restore its KNN in the query range from the optimized ARKGraph index. We conducted extensive experiments on real-world datasets. Experimental results show that our optimized ARKGraph index achieved a small index size, low query latency, and good scalability. Specifically, our approach was 1000x faster than the baseline method that builds a KGraph for all the vectors in the query range on-the-fly.