Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms - an iterative approximation algorithm that converges fast and is accurate, and a subset selection method that makes the computation even faster. It has been previously proposed that magnitude of model sequences generated during stochastic gradient descent is correlated to generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences in fact bear higher correlations. We also describe new applications of magnitude in machine learning - as an effective regularizer for neural network training, and as a novel clustering criterion.
{"title":"Approximating Metric Magnitude of Point Sets","authors":"Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar","doi":"arxiv-2409.04411","DOIUrl":"https://doi.org/arxiv-2409.04411","url":null,"abstract":"Metric magnitude is a measure of the \"size\" of point clouds with many\u0000desirable geometric properties. It has been adapted to various mathematical\u0000contexts and recent work suggests that it can enhance machine learning and\u0000optimization algorithms. But its usability is limited due to the computational\u0000cost when the dataset is large or when the computation must be carried out\u0000repeatedly (e.g. in model training). In this paper, we study the magnitude\u0000computation problem, and show efficient ways of approximating it. We show that\u0000it can be cast as a convex optimization problem, but not as a submodular\u0000optimization. The paper describes two new algorithms - an iterative\u0000approximation algorithm that converges fast and is accurate, and a subset\u0000selection method that makes the computation even faster. It has been previously\u0000proposed that magnitude of model sequences generated during stochastic gradient\u0000descent is correlated to generalization gap. Extension of this result using our\u0000more scalable algorithms shows that longer sequences in fact bear higher\u0000correlations. We also describe new applications of magnitude in machine\u0000learning - as an effective regularizer for neural network training, and as a\u0000novel clustering criterion.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture.
{"title":"Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $mathbb{R}^d$","authors":"Emilio Corso, Pablo Shmerkin","doi":"arxiv-2409.04608","DOIUrl":"https://doi.org/arxiv-2409.04608","url":null,"abstract":"We extend a theorem of the second author on the $L^q$-dimensions of\u0000dynamically driven self-similar measures from the real line to arbitrary\u0000dimension. Our approach provides a novel, simpler proof even in the\u0000one-dimensional case. As consequences, we show that, under mild separation\u0000conditions, the $L^q$-dimensions of homogeneous self-similar measures in\u0000$mathbb{R}^d$ take the expected values, and we derive higher rank slicing\u0000theorems in the spirit of Furstenberg's slicing conjecture.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality for the principal frequency of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995]. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N is close enough to 2 or 3. The inequality contains the so-called ``asymptotic volume ratio", and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the RCD(0,N) space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied.
{"title":"Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability","authors":"Alexandru Kristály, Andrea Mondino","doi":"arxiv-2409.04337","DOIUrl":"https://doi.org/arxiv-2409.04337","url":null,"abstract":"We study fine properties of the principal frequency of clamped plates in the\u0000(possibly singular) setting of metric measure spaces verifying the RCD(0,N)\u0000condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci\u0000curvature and dimension bounded above by N>1 in the synthetic sense. The\u0000initial conjecture -- an isoperimetric inequality for the principal frequency\u0000of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the\u0000Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and\u0000Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995].\u0000The main contribution of the present work is a new isoperimetric inequality for\u0000the principal frequency of clamped plates in RCD(0,N) spaces whenever N is\u0000close enough to 2 or 3. The inequality contains the so-called ``asymptotic\u0000volume ratio\", and turns out to be sharp under the subharmonicity of the\u0000distance function, a condition satisfied in metric measure cones. In addition,\u0000rigidity (i.e., equality in the isoperimetric inequality) and stability results\u0000are established in terms of the cone structure of the RCD(0,N) space as well as\u0000the shape of the eigenfunction for the principal frequency, given by means of\u0000Bessel functions. These results are new even for Riemannian manifolds with\u0000non-negative Ricci curvature. We discuss examples of both smooth and non-smooth\u0000spaces where the results can be applied.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the structure of the minimal displacement set in weakly systolic complexes. We show that such set is systolic and that it embeds isometrically into the complex. As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycentre of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).
{"title":"Minimal displacement set for weakly systolic complexes","authors":"Ioana-Claudia Lazar","doi":"arxiv-2409.03850","DOIUrl":"https://doi.org/arxiv-2409.03850","url":null,"abstract":"We investigate the structure of the minimal displacement set in weakly\u0000systolic complexes. We show that such set is systolic and that it embeds\u0000isometrically into the complex. As corollaries, we prove that any isometry of a\u0000weakly systolic complex either fixes the barycentre of some simplex (elliptic\u0000case) or it stabilizes a thick geodesic (hyperbolic case).","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the question of whether a non-constant absolutely continuous path can be reparametrised as being unit speed with respect to the Kobayashi metric. Even when the answer is "Yes," which isn't always the case, its proof involves some subtleties. We answer the above question and discuss a small application to Kobayashi geometry.
{"title":"Non-smooth paths having unit speed with respect to the Kobayashi metric","authors":"Gautam Bharali, Rumpa Masanta","doi":"arxiv-2409.03709","DOIUrl":"https://doi.org/arxiv-2409.03709","url":null,"abstract":"In this paper, we investigate the question of whether a non-constant\u0000absolutely continuous path can be reparametrised as being unit speed with\u0000respect to the Kobayashi metric. Even when the answer is \"Yes,\" which isn't\u0000always the case, its proof involves some subtleties. We answer the above\u0000question and discuss a small application to Kobayashi geometry.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given $s in (0,1]$ and $t in [0,2]$, suppose a set $X$ in the plane has the following property:~there is a collection of lines of packing dimension $t$ such that every line from the collection intersects $X$ in a set of packing dimension at least $s$. We show that such sets must have packing dimension at least $max{s,t/2}$ and that this bound is sharp. In particular this solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is $max{s,t-1}$.
{"title":"On variants of the Furstenberg set problem","authors":"Jonathan M. Fraser","doi":"arxiv-2409.03678","DOIUrl":"https://doi.org/arxiv-2409.03678","url":null,"abstract":"Given $s in (0,1]$ and $t in [0,2]$, suppose a set $X$ in the plane has the\u0000following property:~there is a collection of lines of packing dimension $t$\u0000such that every line from the collection intersects $X$ in a set of packing\u0000dimension at least $s$. We show that such sets must have packing dimension at\u0000least $max{s,t/2}$ and that this bound is sharp. In particular this solves a\u0000variant of the Furstenberg set problem for packing dimension. We also solve the\u0000upper and lower box dimension variants of the problem. In both of these cases\u0000the sharp threshold is $max{s,t-1}$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-H"older map between shells, and constitutes an addition to a general programme of research proposed by J. Fraser.
{"title":"On concentric fractal spheres and spiral shells","authors":"Efstathios Konstantinos Chrontsios Garitsis","doi":"arxiv-2409.03047","DOIUrl":"https://doi.org/arxiv-2409.03047","url":null,"abstract":"We investigate dimension-theoretic properties of concentric topological\u0000spheres, which are fractal sets emerging both in pure and applied mathematics.\u0000We calculate the box dimension and Assouad spectrum of such collections, and\u0000use them to prove that fractal spheres cannot be shrunk into a point at a\u0000polynomial rate. We also apply these dimension estimates to quasiconformally\u0000classify certain spiral shells, a generalization of planar spirals in higher\u0000dimensions. This classification also provides a bi-H\"older map between shells,\u0000and constitutes an addition to a general programme of research proposed by J.\u0000Fraser.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gromov's Lipschitz order is an order relation on the set of metric measure spaces. One of the compactifications of the space of isomorphism classes of metric measure spaces equipped with the concentration topology is constructed by using the Lipschitz order. The concentration topology is deeply related to the concentration of measure phenomenon. In this paper, we extend the Lipschitz order to that with additive errors and prove useful properties. We also discuss the relation of it to a map with the property of 1-Lipschitz up to an additive error.
{"title":"Extension of Gromov's Lipschitz order to with additive errors","authors":"Hiroki Nakajima","doi":"arxiv-2409.02459","DOIUrl":"https://doi.org/arxiv-2409.02459","url":null,"abstract":"Gromov's Lipschitz order is an order relation on the set of metric measure\u0000spaces. One of the compactifications of the space of isomorphism classes of\u0000metric measure spaces equipped with the concentration topology is constructed\u0000by using the Lipschitz order. The concentration topology is deeply related to\u0000the concentration of measure phenomenon. In this paper, we extend the Lipschitz\u0000order to that with additive errors and prove useful properties. We also discuss\u0000the relation of it to a map with the property of 1-Lipschitz up to an additive\u0000error.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a symbolic classification of generalized Seifert fiber spaces, which were introduced by Mitsuishi and Yamaguchi in the classification of collapsing Alexandrov $3$-spaces. Additionally, we show that the canonical double branched cover of a non-manifold generalized Seifert fiber space is a Seifert manifold and compute its symbolic invariants in terms of those of the original space.
{"title":"Classification of generalized Seifert fiber spaces","authors":"Fernando Galaz-Garcia, Jesús Núñez-Zimbrón","doi":"arxiv-2409.02216","DOIUrl":"https://doi.org/arxiv-2409.02216","url":null,"abstract":"We provide a symbolic classification of generalized Seifert fiber spaces,\u0000which were introduced by Mitsuishi and Yamaguchi in the classification of\u0000collapsing Alexandrov $3$-spaces. Additionally, we show that the canonical\u0000double branched cover of a non-manifold generalized Seifert fiber space is a\u0000Seifert manifold and compute its symbolic invariants in terms of those of the\u0000original space.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An affine P'olya-Szeg"o principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the $L^p$ affine P'olya-Szeg"o principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new P'olya-Szeg"o principle are shown.
{"title":"On the $m$th-order Affine Pólya-Szegö Principle","authors":"Dylan Langharst, Michael Roysdon, Yiming Zhao","doi":"arxiv-2409.02232","DOIUrl":"https://doi.org/arxiv-2409.02232","url":null,"abstract":"An affine P'olya-Szeg\"o principle for a family of affine energies, with\u0000equality condition characterization, is demonstrated. In particular, this\u0000recovers, as special cases, the $L^p$ affine P'olya-Szeg\"o principles due to\u0000Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao.\u0000Various applications of this new P'olya-Szeg\"o principle are shown.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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