150 research outputs found
Randomness extraction and asymptotic Hamming distance
We obtain a non-implication result in the Medvedev degrees by studying
sequences that are close to Martin-L\"of random in asymptotic Hamming distance.
Our result is that the class of stochastically bi-immune sets is not Medvedev
reducible to the class of sets having complex packing dimension 1
Simulation Theorems via Pseudorandom Properties
We generalize the deterministic simulation theorem of Raz and McKenzie
[RM99], to any gadget which satisfies certain hitting property. We prove that
inner-product and gap-Hamming satisfy this property, and as a corollary we
obtain deterministic simulation theorem for these gadgets, where the gadget's
input-size is logarithmic in the input-size of the outer function. This answers
an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result
also implies the previous results for the Indexing gadget, with better
parameters than was previously known. A preliminary version of the results
obtained in this work appeared in [CKL+17]
Indexability, concentration, and VC theory
Degrading performance of indexing schemes for exact similarity search in high
dimensions has long since been linked to histograms of distributions of
distances and other 1-Lipschitz functions getting concentrated. We discuss this
observation in the framework of the phenomenon of concentration of measure on
the structures of high dimension and the Vapnik-Chervonenkis theory of
statistical learning.Comment: 17 pages, final submission to J. Discrete Algorithms (an expanded,
improved and corrected version of the SISAP'2010 invited paper, this e-print,
v3
Efficient sphere-covering and converse measure concentration via generalized coding theorems
Suppose A is a finite set equipped with a probability measure P and let M be
a ``mass'' function on A. We give a probabilistic characterization of the most
efficient way in which A^n can be almost-covered using spheres of a fixed
radius. An almost-covering is a subset C_n of A^n, such that the union of the
spheres centered at the points of C_n has probability close to one with respect
to the product measure P^n. An efficient covering is one with small mass
M^n(C_n); n is typically large. With different choices for M and the geometry
on A our results give various corollaries as special cases, including Shannon's
data compression theorem, a version of Stein's lemma (in hypothesis testing),
and a new converse to some measure concentration inequalities on discrete
spaces. Under mild conditions, we generalize our results to abstract spaces and
non-product measures.Comment: 29 pages. See also http://www.stat.purdue.edu/~yiannis
A Note on the Probability of Rectangles for Correlated Binary Strings
Consider two sequences of independent and identically distributed fair
coin tosses, and , which are
-correlated for each , i.e. .
We study the question of how large (small) the probability can be among all sets of a given cardinality.
For sets it is well known that the largest (smallest)
probability is approximately attained by concentric (anti-concentric) Hamming
balls, and this can be proved via the hypercontractive inequality (reverse
hypercontractivity). Here we consider the case of . By
applying a recent extension of the hypercontractive inequality of
Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming
balls of the same size approximately maximize in
the regime of . We also prove a similar tight lower bound, i.e.
show that for the pair of opposite Hamming balls approximately
minimizes the probability
- β¦