138,136 research outputs found
Monomiality principle, Sheffer-type polynomials and the normal ordering problem
We solve the boson normal ordering problem for
with arbitrary functions and and integer , where and
are boson annihilation and creation operators, satisfying
. This consequently provides the solution for the exponential
generalizing the shift operator. In the
course of these considerations we define and explore the monomiality principle
and find its representations. We exploit the properties of Sheffer-type
polynomials which constitute the inherent structure of this problem. In the end
we give some examples illustrating the utility of the method and point out the
relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics
"Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005),
Myczkowce, Poland. 13 pages, 31 reference
Characterizing the Shape of Activation Space in Deep Neural Networks
The representations learned by deep neural networks are difficult to
interpret in part due to their large parameter space and the complexities
introduced by their multi-layer structure. We introduce a method for computing
persistent homology over the graphical activation structure of neural networks,
which provides access to the task-relevant substructures activated throughout
the network for a given input. This topological perspective provides unique
insights into the distributed representations encoded by neural networks in
terms of the shape of their activation structures. We demonstrate the value of
this approach by showing an alternative explanation for the existence of
adversarial examples. By studying the topology of network activations across
multiple architectures and datasets, we find that adversarial perturbations do
not add activations that target the semantic structure of the adversarial class
as previously hypothesized. Rather, adversarial examples are explainable as
alterations to the dominant activation structures induced by the original
image, suggesting the class representations learned by deep networks are
problematically sparse on the input space
struc2vec: Learning Node Representations from Structural Identity
Structural identity is a concept of symmetry in which network nodes are
identified according to the network structure and their relationship to other
nodes. Structural identity has been studied in theory and practice over the
past decades, but only recently has it been addressed with representational
learning techniques. This work presents struc2vec, a novel and flexible
framework for learning latent representations for the structural identity of
nodes. struc2vec uses a hierarchy to measure node similarity at different
scales, and constructs a multilayer graph to encode structural similarities and
generate structural context for nodes. Numerical experiments indicate that
state-of-the-art techniques for learning node representations fail in capturing
stronger notions of structural identity, while struc2vec exhibits much superior
performance in this task, as it overcomes limitations of prior approaches. As a
consequence, numerical experiments indicate that struc2vec improves performance
on classification tasks that depend more on structural identity.Comment: 10 pages, KDD2017, Research Trac
A Lambda Term Representation Inspired by Linear Ordered Logic
We introduce a new nameless representation of lambda terms inspired by
ordered logic. At a lambda abstraction, number and relative position of all
occurrences of the bound variable are stored, and application carries the
additional information where to cut the variable context into function and
argument part. This way, complete information about free variable occurrence is
available at each subterm without requiring a traversal, and environments can
be kept exact such that they only assign values to variables that actually
occur in the associated term. Our approach avoids space leaks in interpreters
that build function closures.
In this article, we prove correctness of the new representation and present
an experimental evaluation of its performance in a proof checker for the
Edinburgh Logical Framework.
Keywords: representation of binders, explicit substitutions, ordered
contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
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