We extend results of Pachner and Casali to give finite sets of moves relating triangulations of PL manifolds respecting filtrations by locally flat manifolds and stratifications in which a finite family of simple local models exists for neighborhoods of strata.
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We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogu…
The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of isometric embeddings of compact surfaces into . We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, manifolds of dimens…
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be -approximated by PL symplectic manifolds.
Proves common stellar subdivisions for all PL homeomorphic polyhedra.
problem Proving common stellar subdivisions for all PL homeomorphic polyhedra.
method Proved weighted strong factorization conjecture and iterated blowups.
result Every two PL homeomorphic polyhedra have a common stellar subdivision.
This study examines the relationship between PLS and OLS regression using eigenvalue distributions.
problem Analyzing the difference between PLS and OLS regression in terms of eigenvalue distributions.
method Examined the distance between PLS and OLS regression coefficients using the Mahalanobis distance and eigenvalue distributions of the regressor covariance matrix.
result Provided a bound on the distance between PLS and OLS regression coefficients that depends only on the eigenvalue distribution of the regressor covariance matrix.
After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, an…
For , the regular genus of a closed connected PL -manifold is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…
We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL collapsible subpolyhedra in terms of their simple homotopy type and…
Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…
New method speeds up NIR spectroscopy calibration by 400x.
problem Efficient preprocessing selection in NIR spectroscopy.
method Operator-adaptive PLS and Ridge regression.
result Significant reduction in fitting time with comparable prediction quality.
New 5-manifold without 'spine' challenges deformation conjecture.
problem Existence of exotic smooth structures on 4-manifolds.
method Constructed a compact PL 5-manifold homotopy equivalent to a wedge of 11 spheres.
result The constructed manifold is 'spineless', not a regular neighborhood of any 2-complex.
A space is created to realize a specific cohomology module, showing PL structure but not smoothability.
problem Realizing a specific cohomology module as the cohomology of a space.
method Obstruction theory and attaching extra cells to create a Poincaré duality space.
result The space admits a PL structure but not smoothability.
The paper classifies certain PL manifolds using PL cobordism.
problem Classifying PL manifolds of specific dimensions.
method PL cobordism theory and computations of bordism groups.
result Classification of PL manifolds in dimensions 10 and 13.
Preference learning (PL) is a core area of machine learning that handles datasets with ordinal relations. As the number of generated data of ordinal nature is increasing, the importance and role of the PL field becomes central within machine learning research and practice. This paper introduces an open source, scalable…
A classical result by Pachner states that two -dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly -colored, manifolds such a sequence can be chosen such that bala…
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
PLS-Lasso integrates dimension reduction into regression for financial index tracking.
problem Dimension reduction and regression are traditionally treated separately in multivariate data analysis.
method PLS-Lasso integrates dimension reduction directly into the regression process, presenting two formulations: PLS-Lasso-v1 and PLS-Lasso-v2.
result PLS-Lasso-v1 and PLS-Lasso-v2 outperform Lasso in financial index tracking.
New PL invariant classifies K3 surface degenerations.
problem Classifying type II degenerations of K3 surfaces.
method Explicit PL convex function from interval, differential geometric viewpoint.
result Function classifies degenerations into combinatorial types.
Proves PL cobordism category's homotopy type, analogous to smooth case.
problem Describes the homotopy type of PL cobordism category.
method Introduces PL Madsen-Tillmann spectrum and proves weak equivalence.
result Weak homotopy equivalence between PL cobordism category and Ω^∞-1 MTPL(d).
This work proposes a mathematical framework for loss landscapes and optimization in deep neural networks.
problem The effectiveness of gradient-based optimization in over-parameterized neural networks.
method A modern view and mathematical framework of loss landscapes and efficient optimization in over-parameterized machine learning models.
result Wide neural networks satisfy the PL condition, explaining (S)GD convergence to a global minimum.
Paper confirms conjecture for PL foliations of codimension 2.
problem Confirming Haefliger-Thurston's conjecture for PL foliations.
method Using a version of Mather-Thurston's theorem for PL homeomorphisms.
result Derive new homological properties for PL surface homeomorphisms.
Suppose M is a noncompact connected PL 2-manifold. In this paper we study the topological property of the triple (H(M)_0, H^PL(M)_0, H^PL, c(M)_0), where H(M)_0 is the identity component of the homeomorphism group {\cal H}(M) of M with the compact-open topology, and H^PL(M)_0 and H^PL, c(M)_0 are the identity component…
The study explores convex unions and completions in simplicial pseudomanifolds, revealing unexpected behavior.
problem Understanding the behavior of convex unions in simplicial pseudomanifolds.
method Generalization to simplicial pseudomanifolds, considering PL homeomorphisms and edge subdivisions.
result Unexpected behavior in convex unions and completions, including empty contraction spaces and large/small contraction spaces.
High-dimensional data common in genomics, proteomics, and chemometrics often contains complicated correlation structures. Recently, partial least squares (PLS) and Sparse PLS methods have gained attention in these areas as dimension reduction techniques in the context of supervised data analysis. We introduce a framewo…
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
A new PMA method combines PCA and PLS for better classification.
problem Improving classification performance in data analysis.
method Combines PCA and PLS through multiple sub-PLS models and PCA on the joint coefficient matrix.
result The proposed PMA method achieves better classification performance and stability.
There are 2^n possible resolutions of a smooth pseudodiagram with n precrossings. If we consider piecewise-linear (PL) pseudodiagrams and resolutions that themselves are PL, certain resolutions of the pseudodiagram may not exist in three-space. We investigate this situation and its impact on the weighted resolution set…
A new method for fair representation learning using PLS.
problem Fairness in representation learning for data reduction.
method Proposes Fair Partial Least Squares (PLS) components with fairness constraints.
result The new method outperforms standard fair PCA methods on various datasets.
Stochastic gradient methods are dominant in nonconvex optimization especially for deep models but have low asymptotical convergence due to the fixed smoothness. To address this problem, we propose a simple yet effective method for improving stochastic gradient methods named predictive local smoothness (PLS). First, we …
Simplified proof of Lefschetz theorem for PL spheres.
problem Proving the Lefschetz theorem for PL spheres.
method Using Pachner's Theorem to replace geometric constructions.
result Simpler proof and implications for g-conjecture and Kalai-Sarkaria conjecture.
Quantitative PL bordismmath.GT
Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
Open and discrete maps with specific branch set images are equivalent to PL branched covers.
problem Understanding the equivalence of open and discrete maps and PL branched covers.
method Demonstrated that an open and discrete map with a specific branch set image is equivalent to a PL branched cover up to homeomorphism.
result Open and discrete maps with a specific branch set image are equivalent to PL branched covers.
The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the to…
The regular genus of certain 4-manifolds is determined, providing new insights.
problem Determining the regular genus of higher-dimensional closed PL manifolds.
method Using crystallization graphs and combinatorial topology, the regular genus is calculated for specific manifolds.
result The regular genus of is 6, and is 16.
Study PL topology theorems for cubical complexes, solving Habegger and Funar's conjecture.
problem Characterize PL homeomorphic cubulations equivalence by Pachner moves.
method Show equivalence to the existence of cobordisms between generic immersions of hypersurfaces.
result Solve Habegger and Funar's conjecture about PL homeomorphic cubulations equivalence.
Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible…
Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problemsmath.GT
Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
PL Morse theory proves strong regularity in low dimensions.
problem Understanding regular and critical points in PL manifolds.
method Introducing homologically and strongly regular points, presenting criteria, and constructing examples.
result In low dimensions , homologically regular points are always strongly regular.
Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any (simply-connected) PL -manifold admitting a simple crystallization admits a special hand…
R-PLS improves analysis of brain functional connectivity matrices.
problem Improving analysis of functional connectivity matrices in brain imaging.
method Introducing R-PLS, a generalization of PLS for symmetric positive definite matrices.
result R-PLS identifies key functional connections in brain imaging datasets.
Bayesian Pseudo Label Selection reduces overfitting in semi-supervised learning.
problem Overfitting in pseudo-label selection for semi-supervised learning.
method BPLS, a Bayesian framework that approximates the posterior predictive of pseudo-samples.
result BPLS outperforms traditional PLS methods, especially in high-dimensional data.
Bing's house-like spines approximate all PL manifolds.
problem Understanding spines of PL manifolds.
method Approximation rigidity and h-principle.
result PL manifolds have spines similar to Bing's house.
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
PL-MCMC samples from normalizing flows' conditional distributions.
problem Sampling from complex conditional distributions learned by normalizing flows.
method Metropolis-Hastings implementation of PL-MCMC.
result PL-MCMC asymptotically samples from exact conditional distributions.
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
Let M be a PL 2-manifold and X be a compact subpolyhedron of M and let E(X, M) denote the space of embeddings of X into M with the compact-open topology. In this paper we study an extension property of embeddings of X into M and show that the restriction map from the homeomorphism group of M to E(X, M) is a principal b…