34 research outputs found
Algorithmic statistics revisited
The mission of statistics is to provide adequate statistical hypotheses
(models) for observed data. But what is an "adequate" model? To answer this
question, one needs to use the notions of algorithmic information theory. It
turns out that for every data string one can naturally define
"stochasticity profile", a curve that represents a trade-off between complexity
of a model and its adequacy. This curve has four different equivalent
definitions in terms of (1)~randomness deficiency, (2)~minimal description
length, (3)~position in the lists of simple strings and (4)~Kolmogorov
complexity with decompression time bounded by busy beaver function. We present
a survey of the corresponding definitions and results relating them to each
other
Algorithmic statistics: forty years later
Algorithmic statistics has two different (and almost orthogonal) motivations.
From the philosophical point of view, it tries to formalize how the statistics
works and why some statistical models are better than others. After this notion
of a "good model" is introduced, a natural question arises: it is possible that
for some piece of data there is no good model? If yes, how often these bad
("non-stochastic") data appear "in real life"?
Another, more technical motivation comes from algorithmic information theory.
In this theory a notion of complexity of a finite object (=amount of
information in this object) is introduced; it assigns to every object some
number, called its algorithmic complexity (or Kolmogorov complexity).
Algorithmic statistic provides a more fine-grained classification: for each
finite object some curve is defined that characterizes its behavior. It turns
out that several different definitions give (approximately) the same curve.
In this survey we try to provide an exposition of the main results in the
field (including full proofs for the most important ones), as well as some
historical comments. We assume that the reader is familiar with the main
notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde
On Algorithmic Statistics for space-bounded algorithms
Algorithmic statistics studies explanations of observed data that are good in
the algorithmic sense: an explanation should be simple i.e. should have small
Kolmogorov complexity and capture all the algorithmically discoverable
regularities in the data. However this idea can not be used in practice because
Kolmogorov complexity is not computable.
In this paper we develop algorithmic statistics using space-bounded
Kolmogorov complexity. We prove an analogue of one of the main result of
`classic' algorithmic statistics (about the connection between optimality and
randomness deficiences). The main tool of our proof is the Nisan-Wigderson
generator.Comment: accepted to CSR 2017 conferenc
Universal equivalence and majority of probabilistic programs over finite fields
We study decidability problems for equivalence of probabilistic programs, for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication and conditionals and thus is sufficiently expressive to encode boolean and arithmetic circuits. We consider two variants of equivalence: the first one considers an interpretation over a fixed finite field, while the second one, which we call universal equivalence, verifies equivalence over all extensions of a finite field. The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of extensions of the boolean field. While the first problem is obviously decidable, we establish its exact complexity which lies in the counting hierarchy. To show decidability, and a doubly exponential upper bound, of the universal variant we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. Finally we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy
Reachability Problems in Nondeterministic Polynomial Maps on the Integers
We study the reachability problems in various nondeterministic
polynomial maps in Zn. We prove that the reachability problem for
very simple three-dimensional affine maps (with independent variables)
is undecidable and is PSPACE-hard for two-dimensional quadratic maps.
Then we show that the complexity of the reachability problem for maps
without functions of the form ±x + b is lower. In this case the reachability
problem is PSPACE-complete in general, and NP-hard for any fixed
dimension. Finally we extend the model by considering maps as language
acceptors and prove that the universality problem is undecidable
for two-dimensional affine maps
RECURSIA-RRT:Recursive translatable point-set pattern discovery with removal of redundant translators
We introduce two algorithms, RECURSIA and RRT, designed to increase the
compression factor achievable using point-set cover algorithms based on the SIA
and SIATEC pattern discovery algorithms. SIA computes the maximal translatable
patterns (MTPs) in a point set, while SIATEC computes the translational
equivalence class (TEC) of every MTP in a point set, where the TEC of an MTP is
the set of translationally invariant occurrences of that MTP in the point set.
In its output, SIATEC encodes each MTP TEC as a pair, , where P is the
first occurrence of the MTP and V is the set of non-zero vectors that map P
onto its other occurrences. RECURSIA recursively applies a TEC cover algorithm
to the pattern P, in each TEC, , that it discovers. RRT attempts to remove
translators from V in each TEC without reducing the total set of points covered
by the TEC. When evaluated with COSIATEC, SIATECCompress and Forth's algorithm
on the JKU Patterns Development Database, using RECURSIA with or without RRT
increased compression factor and recall but reduced precision. Using RRT alone
increased compression factor and reduced recall and precision, but had a
smaller effect than RECURSIA.Comment: Submitted to 12th International Workshop on Machine Learning and
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