282 research outputs found
On monotone circuits with local oracles and clique lower bounds
We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , under a certain restrictive
assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles,
extends and provides a matching upper bound for the exponential lower bounds on
the monotone circuit size complexity of -clique obtained by Alon and Boppana
(1987).Comment: Updated acknowledgements and funding informatio
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
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