UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology The result is a practical scalable algorithm that applies to real world data The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning

UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology It is also clear that the nature of the data we are obtaining is significantly different For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it The data obtained is also often much noisier than in the past and has more missing information (missing data) This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, ie finite sets of points equipped with a distance function It then adapts tools from the various branches of geometry to the study of point clouds The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise Here are some of the key points which come up when applying these geometric methods to data analysis • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, ie to understand how it is organized on a large scale For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key

Topology and data

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

Higher Topos Theory

An introduction to the homotopy groups of spheres Setting up the Adams spectral sequence The classical Adams spectral sequence $BP$-theory and the Adams-Novikov spectral sequence The chromatic spectral sequence Morava stabilizer algebras Computing stable homotopy groups with the Adams-Novikov spectral sequence Hopf algebras and Hopf algebroids Formal group laws Tables of homotopy groups of spheres Bibliography Index

Complex Cobordism and Stable Homotopy Groups of Spheres

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K  S1 to be a torus. Invent. math. 131, 25±83 (1998)

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Equivariant cohomology, Koszul duality, and the localization theorem

Simplicial sets.- Model Categories.- Classical results and constructions.- Bisimplicial sets.- Simplicial groups.- The homotopy theory of towers.- Reedy model categories.- Cosimplicial spaces: applications.- Simplicial functors and homotopy coherence.- Localization.

Simplicial homotopy theory

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n =2a tp = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre’s computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel’s nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the “chromatic” point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings – most notably by Shimomura and his coworkers, and Ravenel, and more lately by Hopkins and

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A resolution of the K(2)-local sphere at the prime 3

Simplicial Homotopy Theory: Progress in Mathematics 174

Preface. Conductors in the Non-separable Residue Field Case R. Boltje, G.-M. Cram, V.P. Snaith. On the Reciprocity Sequence in the Higher Class Field Theory of Function Fields J.-L. Colliot-Thelene. Resultats de 'purete' pour les varietes lisses sur un corps fini: Appendice a l'article de J.-L. Colliot-Thelene B. Kahn. Module Structures on the Hochschild and Cyclic Homology of Graded Rings B.H. Dayton, C.A. Weibel. The Intersection Homology and Derived Category of Algebraic Stacks R. Joshua. On the Lichtenbaum--Quillen Conjecture B. Kahn. Tate Motives and the Vanishing Conjectures for Algebraic K-Theory M. Levine. Suslin Homology and Deligne 1-Motives S. Lichtenbaum. On p-adic Topological K-Theory S.A. Mitchell. Life after the Telescope Conjecture D.C. Ravenel. Finiteness of Subintegrality L. Reid, L.G. Roberts, B. Singh. Cohomological Hasse Principle for a Threefold over a Finite Field S. Saito. Les K-groupes d'un fibre projectif R.W. Thomason. Etale Chern Classes at the Prime 2 C. Weibel. Cosimplicial Objects in Algebraic Geometry Z. Wojtkowiak.

Algebraic K-Theory and Algebraic Topology

There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras, it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well.

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Paul G. Goerss

Papers

Simplicial homotopy theory

A resolution of the K(2)-local sphere at the prime 3

Simplicial Homotopy Theory: Progress in Mathematics 174

Algebraic K-Theory and Algebraic Topology

Model Categories and Simplicial Methods