Limitations of Learning via Embeddings in Euclidean Half-Spaces

Ben-David, Shai; Eiron, Nadav; Simon, Hans Ulrich

doi:10.1007/3-540-44581-1_25

Cited by 48 publications

(64 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [4], it was shown that the margin complexity of a sign matrix is almost always Ω( n/ log n). Here we improve this result and show that the margin complexity and γ 2 of sign matrices are almost always Θ( √ n).…”

Section: Typical Values and Concentration Of Measurementioning

confidence: 99%

Complexity measures of sign matrices

et al. 2007

View full text Add to dashboard Cite

show abstract

Section: Typical Values and Concentration Of Measurementioning

confidence: 99%

Complexity measures of sign matrices

et al. 2007

View full text Add to dashboard Cite

show abstract

“…Similarly to the sign rank, define γ ∞ 2 (A) as the minimum γ 2 norm of a matrix that has the same sign pattern as A and all entries at least 1 in magnitude. It is known that rank ± (A) = O(γ ∞ 2 (A) log(mn)) [12] and also that the sign rank can be exponentially smaller than γ ∞ 2 [14,40]. Linial and Shraibman show that γ ∞ 2 (A G ) = Ω( √ d) for a random d-regular graph, which also implies an Ω(d) lower bound on the -approximate rank for any constant < 1/2.…”

Section: Resultsmentioning

confidence: 99%

The approximate rank of a matrix and its algorithmic applications

Alon

Lee

Shraibman

et al. 2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

We study the -rank of a real matrix A, defined for any > 0 as the minimum rank over matrices that approximate every entry of A to within an additive . This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the -rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive approximation schemes for Nash equilibria for 2-player games when the payoff matrices are positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an efficient algorithm for the following problem: given a convex body A and a symmetric convex body B, find a covering a A with translates of B.

show abstract

“…In these cases however, as the dimension of the feature space associated with the kernel becomes exponentially larger, there is an increasing probability that a significant fraction of the feature space dimensions will be poorly correlated with the target function. As a consequence, even when using large margin classifiers, one can fail to obtain models with good generalization performance (Ben-David et al 2002). In order to tackle these issues, several remedies have been proposed, from down-weighting the contribution of larger fragments and/or bounding a priori their size, to a direct manipulation of the Gram matrix.…”

Section: Related Workmentioning

confidence: 99%

Effective feature construction by maximum common subgraph sampling

et al. 2010

View full text Add to dashboard Cite

The standard approach to feature construction and predictive learning in molecular datasets is to employ computationally expensive graph mining techniques and to bias the feature search exploration using frequency or correlation measures. These features are then typically employed in predictive models that can be constructed using, for example, SVMs or decision trees. We take a different approach: rather than mining for all optimal local patterns, we extract features from the set of pairwise maximum common subgraphs. The maximum common subgraphs are computed under the block-and-bridge-preserving subgraph isomorphism from the outerplanar examples in polynomial time. We empirically observe a significant increase in predictive performance when using maximum common subgraph features instead of correlated local patterns on 60 benchmark datasets from NCI. Moreover, we show that when we randomly sample the pairs of graphs from which to extract the maximum common subgraphs, we obtain a smaller set of features that still allows the same predictive performance as methods that exhaustively enumerate all possible patterns. The sampling strategy turns out to be a very good compromise between a slight decrease in predictive performance (although still remaining comparable with state-of-the-art methods) and a significant runtime reduction (two orders of magnitude on a popular medium size chemoinformatics dataset). This suggests that maximum common subgraphs are interesting and meaningful features.

show abstract

Limitations of Learning via Embeddings in Euclidean Half-Spaces

Cited by 48 publications

References 6 publications

Complexity measures of sign matrices

Complexity measures of sign matrices

The approximate rank of a matrix and its algorithmic applications

Effective feature construction by maximum common subgraph sampling

Contact Info

Product

Resources

About