This is the supplementary material for the paper entitled “Deep Adaptive Image Clustering”. The supplementary material is organized as follows. Section 1 gives the mapping function described in Figure 1. Section 2 presents the proof of Theorem 1. Section 3 details the experimental settings in our experiments. 1. The Mapping Function Utilized in Figure 1 We assume that li represents the label fe...

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a non-separation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fac...

This article, which is an accompanying paper to [BLS09], consists of two parts: In section 2 we present a version of Fenchel’s perturbation method for the duality theory of the Monge– Kantorovich problem of optimal transport. The treatment is elementary as we suppose that the spaces (X,μ), (Y, ν), on which the optimal transport problem [Vil03, Vil09] is defined, simply equal the finite set {1, ...

Bump mapping is a normal-perturbation rendering technique for simulating lighting effects caused by patterned irregularities on otherwise locally smooth surfaces. By encoding such surface patterns in texture maps, texture-based bump mapping simulates a surface’s irregular lighting appearance without modeling the patterns as true geometric perturbations to the surface. Bump mapping is advantageo...

We take the base field to be the field of complex numbers in these lectures. The varieties are, by definition, quasi-projective, reduced (but not necessarily irreducible) schemes. Let G be a semisimple, simply-connected, complex algebraic group with a fixed Borel subgroup B, a maximal torus H ⊂ B, and associated Weyl group W . (Recall that a Borel subgroup is any maximal connected, solvable sub...

An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations En where all En-equivalence classs are finite. In this article we establish the following theorem: if a countable abelian group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructio...

Given a measurable mapping f from a nonatomic Loeb probability space (T; T ; P ) to the space of Borel probability measures on a compact metric space A, we show the existence of a measurable mapping g from (T; T ; P ) to A itself such that f and g yield the same values for the integrals associated with a countable class of functions on T A. A corollary generalizes the classical result of Dvoret...

A topological space is a Borel space if it is homeomorphic to a Borel subset of a Polish space (Bertsekas and Shreve, 1978, Definition 7.7). Examples of Borel spaces include any Borel subset of a Euclidean space Rd and, more generally, any Borel subset of a Polish space (Bertsekas and Shreve, 1978, Proposition 7.12). If X and Y are Borel spaces, a function g : X × Y → R is upper semianalytic if...

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel c...