1,080 research outputs found
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball
We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming
ball of equal volume. More precisely, we show that for all even n there exists
an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball
of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x
and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y)
where distance(,) denotes the Hamming distance. In particular, this implies
that the Hamming ball is bi-Lipschitz transitive.
This result gives a strong negative answer to an open problem of Lovett and
Viola [CC 2012], who raised the question in the context of sampling
distributions in low-level complexity classes. The conceptual implication is
that the problem of proving lower bounds in the context of sampling
distributions will require some new ideas beyond the sensitivity-based
structural results of Boppana [IPL 97].
We study the mapping f further and show that it (and its inverse) are
computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f
is "approximately local" in the sense that all but the last output bit of f are
essentially determined by a single input bit
A Hierarchy of Maps Between Compacta
Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair of subclasses of CH, we define Lev≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k \u3c ω and Lev≥ k(K,K) = Lev≥ k+1 (K,K), then Lev≥ k(K,K) = Lev≥ω(K,K). A space X ∈ K is co-existentially closed in K if Lev≥ 0(K, X) = Lev≥ 1 (K, X). Adapting the technique of adding roots, by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension on
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
Weak embeddings of posets to the Boolean lattice
The goal of this paper is to prove that several variants of deciding whether
a poset can be (weakly) embedded into a small Boolean lattice, or to a few
consecutive levels of a Boolean lattice, are NP-complete, answering a question
of Griggs and of Patkos. As an equivalent reformulation of one of these
problems, we also derive that it is NP-complete to decide whether a given graph
can be embedded to the two middle levels of some hypercube
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