141,541 research outputs found
On the Sample Complexity of Predictive Sparse Coding
The goal of predictive sparse coding is to learn a representation of examples
as sparse linear combinations of elements from a dictionary, such that a
learned hypothesis linear in the new representation performs well on a
predictive task. Predictive sparse coding algorithms recently have demonstrated
impressive performance on a variety of supervised tasks, but their
generalization properties have not been studied. We establish the first
generalization error bounds for predictive sparse coding, covering two
settings: 1) the overcomplete setting, where the number of features k exceeds
the original dimensionality d; and 2) the high or infinite-dimensional setting,
where only dimension-free bounds are useful. Both learning bounds intimately
depend on stability properties of the learned sparse encoder, as measured on
the training sample. Consequently, we first present a fundamental stability
result for the LASSO, a result characterizing the stability of the sparse codes
with respect to perturbations to the dictionary. In the overcomplete setting,
we present an estimation error bound that decays as \tilde{O}(sqrt(d k/m)) with
respect to d and k. In the high or infinite-dimensional setting, we show a
dimension-free bound that is \tilde{O}(sqrt(k^2 s / m)) with respect to k and
s, where s is an upper bound on the number of non-zeros in the sparse code for
any training data point.Comment: Sparse Coding Stability Theorem from version 1 has been relaxed
considerably using a new notion of coding margin. Old Sparse Coding Stability
Theorem still in new version, now as Theorem 2. Presentation of all proofs
simplified/improved considerably. Paper reorganized. Empirical analysis
showing new coding margin is non-trivial on real dataset
A variational problem for multifunctions with interaction between leaves
We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions
Relaxed Three-Algebras: Their Matrix Representations and Implications for Multi M2-brane Theory
We argue that one can relax the requirements of the non-associative
three-algebras recently used in constructing D=3, N=8 superconformal field
theories, and introduce the notion of ``relaxed three-algebras''. We present a
specific realization of the relaxed three-algebras in terms of classical Lie
algebras with a matrix representation, endowed with a non-associative
four-bracket structure which is prescribed to replace the three-brackets of the
three-algebras. We show that both the so(4)-based solutions as well as the
cases with non-positive definite metric find a uniform description in our
setting. We discuss the implications of our four-bracket representation for the
D=3, N=8 and multi M2-brane theory and show that our setup can shed light on
the problem of negative kinetic energy degrees of freedom of the Lorentzian
case.Comment: 31 pages, no figure
Manifestly supersymmetric M-theory
In this paper, the low-energy effective dynamics of M-theory,
eleven-dimensional supergravity, is taken off-shell in a manifestly
supersymmetric formulation. We show that a previously proposed relaxation of
the superspace torsion constraints does indeed accommodate a current
supermultiplet which lifts the equations of motion corresponding to the
ordinary second order derivative supergravity lagrangian. Whether the auxiliary
fields obtained this way can be used to construct an off-shell lagrangian is
not yet known. We comment on the relation and application of this completely
general formalism to higher-derivative (R^4) corrections. Some details of the
calculation are saved for a later publication.Comment: 13 pages, plain tex. v2: minor changes, one ref. adde
- …