229,660 research outputs found
The Dimensions of Individual Strings and Sequences
A constructive version of Hausdorff dimension is developed using constructive
supergales, which are betting strategies that generalize the constructive
supermartingales used in the theory of individual random sequences. This
constructive dimension is used to assign every individual (infinite, binary)
sequence S a dimension, which is a real number dim(S) in the interval [0,1].
Sequences that are random (in the sense of Martin-Lof) have dimension 1, while
sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is
shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is
a \Delta^0_2 sequence S such that \dim(S) = \alpha.
A discrete version of constructive dimension is also developed using
termgales, which are supergale-like functions that bet on the terminations of
(finite, binary) strings as well as on their successive bits. This discrete
dimension is used to assign each individual string w a dimension, which is a
nonnegative real number dim(w). The dimension of a sequence is shown to be the
limit infimum of the dimensions of its prefixes.
The Kolmogorov complexity of a string is proven to be the product of its
length and its dimension. This gives a new characterization of algorithmic
information and a new proof of Mayordomo's recent theorem stating that the
dimension of a sequence is the limit infimum of the average Kolmogorov
complexity of its first n bits.
Every sequence that is random relative to any computable sequence of
coin-toss biases that converge to a real number \beta in (0,1) is shown to have
dimension \H(\beta), the binary entropy of \beta.Comment: 31 page
Playing off-line games with bounded rationality
We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in
pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at
most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in
pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at
most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.Refereed Working Papers / of international relevanc
Compressing Sparse Sequences under Local Decodability Constraints
We consider a variable-length source coding problem subject to local
decodability constraints. In particular, we investigate the blocklength scaling
behavior attainable by encodings of -sparse binary sequences, under the
constraint that any source bit can be correctly decoded upon probing at most
codeword bits. We consider both adaptive and non-adaptive access models,
and derive upper and lower bounds that often coincide up to constant factors.
Notably, such a characterization for the fixed-blocklength analog of our
problem remains unknown, despite considerable research over the last three
decades. Connections to communication complexity are also briefly discussed.Comment: 8 pages, 1 figure. First five pages to appear in 2015 International
Symposium on Information Theory. This version contains supplementary materia
Motion characterization from temporal cooccurrences of local motion-based measures for video indexing
This paper describes an original approach for motion interpretation with a view to content-based video indexing. We exploit a statistical analysis of the temporal distribution of appropriate local motion-based measures to perform a global motion characterization. We consider motion features extracted from temporal cooccurrence matrices, and related to properties of homogeneity, acceleration or complexity. Results on various real video sequences are reported and provide a first validation of the approach. 1
Approximate F_2-Sketching of Valuation Functions
We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates.
Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting
Diversity of the G-protein family: sequences from five additional α subunits in the mouse
Biochemical analysis has revealed a number of guanine nucleotide-binding regulatory proteins (G proteins) that mediate signal transduction in mammalian systems. Characterization of their cDNAs uncovered a family of proteins with regions of highly conserved amino acid sequence. To examine the extent of diversity of the G protein family, we used the polymerase chain reaction to detect additional gene products in mouse brain and spermatid RNA that share these conserved regions. Sequences corresponding to six of the eight known G protein alpha subunits were obtained. In addition, we found sequences corresponding to five newly discovered alpha subunits. Our results suggest that the complexity of the G protein family is much greater than previously suspected
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