402,923 research outputs found
Classifying the Arithmetical Complexity of Teaching Models
This paper classifies the complexity of various teaching models by their
position in the arithmetical hierarchy. In particular, we determine the
arithmetical complexity of the index sets of the following classes: (1) the
class of uniformly r.e. families with finite teaching dimension, and (2) the
class of uniformly r.e. families with finite positive recursive teaching
dimension witnessed by a uniformly r.e. teaching sequence. We also derive the
arithmetical complexity of several other decision problems in teaching, such as
the problem of deciding, given an effective coding of all uniformly r.e. families, any such that
, any and , whether or not the
teaching dimension of with respect to is upper bounded
by .Comment: 15 pages in International Conference on Algorithmic Learning Theory,
201
Martingale families and dimension in P
AbstractWe introduce a new measure notion on small complexity classes (called F-measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. On larger complexity classes (E and above), F-measure is equivalent to Lutz resource-bounded measure. As applications to F-measure, we answer a question raised in [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818] by improving their result to: for almost every language A decidable in subexponential time, PA=BPPA. We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F-measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [J.H. Lutz, Dimension in complexity classes, in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158–169] on P, which meets the intuition behind Lutz’s notion. We show that P-dimension lies between finite-state dimension and dimension on E. We prove an analogue of a Theorem of Eggleston in P, i.e. the class of languages whose characteristic sequence contains 1’s with frequency α, has dimension the Shannon entropy of α in P
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
Representation Learning for Clustering: A Statistical Framework
We address the problem of communicating domain knowledge from a user to the
designer of a clustering algorithm. We propose a protocol in which the user
provides a clustering of a relatively small random sample of a data set. The
algorithm designer then uses that sample to come up with a data representation
under which -means clustering results in a clustering (of the full data set)
that is aligned with the user's clustering. We provide a formal statistical
model for analyzing the sample complexity of learning a clustering
representation with this paradigm. We then introduce a notion of capacity of a
class of possible representations, in the spirit of the VC-dimension, showing
that classes of representations that have finite such dimension can be
successfully learned with sample size error bounds, and end our discussion with
an analysis of that dimension for classes of representations induced by linear
embeddings.Comment: To be published in Proceedings of UAI 201
- …