4,849 research outputs found
Axiomatic Characterization of Data-Driven Influence Measures for Classification
We study the following problem: given a labeled dataset and a specific
datapoint x, how did the i-th feature influence the classification for x? We
identify a family of numerical influence measures - functions that, given a
datapoint x, assign a numeric value phi_i(x) to every feature i, corresponding
to how altering i's value would influence the outcome for x. This family, which
we term monotone influence measures (MIM), is uniquely derived from a set of
desirable properties, or axioms. The MIM family constitutes a provably sound
methodology for measuring feature influence in classification domains; the
values generated by MIM are based on the dataset alone, and do not make any
queries to the classifier. While this requirement naturally limits the scope of
our framework, we demonstrate its effectiveness on data
Influence in Classification via Cooperative Game Theory
A dataset has been classified by some unknown classifier into two types of
points. What were the most important factors in determining the classification
outcome? In this work, we employ an axiomatic approach in order to uniquely
characterize an influence measure: a function that, given a set of classified
points, outputs a value for each feature corresponding to its influence in
determining the classification outcome. We show that our influence measure
takes on an intuitive form when the unknown classifier is linear. Finally, we
employ our influence measure in order to analyze the effects of user profiling
on Google's online display advertising.Comment: accepted to IJCAI 201
Altitude Training: Strong Bounds for Single-Layer Dropout
Dropout training, originally designed for deep neural networks, has been
successful on high-dimensional single-layer natural language tasks. This paper
proposes a theoretical explanation for this phenomenon: we show that, under a
generative Poisson topic model with long documents, dropout training improves
the exponent in the generalization bound for empirical risk minimization.
Dropout achieves this gain much like a marathon runner who practices at
altitude: once a classifier learns to perform reasonably well on training
examples that have been artificially corrupted by dropout, it will do very well
on the uncorrupted test set. We also show that, under similar conditions,
dropout preserves the Bayes decision boundary and should therefore induce
minimal bias in high dimensions.Comment: Advances in Neural Information Processing Systems (NIPS), 201
The Dirichlet Problem for Harmonic Functions on Compact Sets
For any compact set we develop the theory of Jensen
measures and subharmonic peak points, which form the set , to
study the Dirichlet problem on . Initially we consider the space of
functions on which can be uniformly approximated by functions harmonic in a
neighborhood of as possible solutions. As in the classical theory, our
Theorem 8.1 shows for compact sets with
closed. However, in general a continuous solution cannot be
expected even for continuous data on \rO_K as illustrated by Theorem 8.1.
Consequently, we show that the solution can be found in a class of finely
harmonic functions. Moreover by Theorem 8.7, in complete analogy with the
classical situation, this class is isometrically isomorphic to
for all compact sets .Comment: There have been a large number of changes made from the first
version. They mostly consists of shortening the article and supplying
additional reference
Partial separability revisited: Necessary and sufficient criteria
We extend the classification of mixed states of quantum systems composed of
arbitrary number of subsystems of arbitrary dimensions. This extended
classification is complete in the sense of partial separability and gives
1+18+1 partial separability classes in the tripartite case contrary to a former
1+8+1. Then we give necessary and sufficient criteria for these classes, which
make it possible to determine to which class a mixed state belongs. These
criteria are given by convex roof extensions of functions defined on pure
states. In the special case of three-qubit systems, we define a different set
of such functions with the help of the Freudenthal triple system approach of
three-qubit entanglement.Comment: v3: 22 pages, 5 tables, 1 figure, minor corrections (typos),
clarification in the Introduction. Accepted in Phys. Rev. A. Comments are
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Uncertainty in Economic Growth and Inequality
A step to consilience, starting with a deconstruction of the causality of
uncertainty that is embedded in the fundamentals of growth and inequality,
following a construction of aggregation laws that disclose the invariance
principle across heterogeneous individuals, ending with a reconstruction of
metric models that yields deeper structural connections via U.S. GDP and income
data
Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity
We introduce two new "degree of complementarity" measures: supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with foundations - i.e. level 0 of the hierarchies - resting exactly on submodular and subadditive functions, respectively.
We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy: Every monotone set function of supermodular degree d has supermodular width at most d, and there exists a supermodular-width-1 function over a ground set of m elements whose supermodular degree is m-1. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the supermodular degree to the supermodular width. We also establish almost matching information-theoretical lower bounds for these two well-studied fundamental maximization problems over set functions. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides not only a natural notion of complementarity, but also an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We also provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy
New tools for dealing with errors-in-variables in DEA.
Errors in variables; Tool;
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