8,947 research outputs found
Resolutions for principal series representations of p-adic GL(n)
Let F be a nonarchimedean locally compact field with residue characteristic p
and G(F) the group of F-rational points of a connected reductive group.
Following Schneider and Stuhler, one can realize, in a functorial way, any
smooth complex finitely generated representation of G(F) as the 0-homology of a
certain coefficient system on the semi-simple building of G(F). It is known
that this method does not apply in general for smooth mod p representations of
G(F), even when G= GL(2). However, we prove that a principal series
representation of GL(n,F) over a field with arbitrary characteristic can be
realized as the 0-homology of the corresponding coefficient system
Auto-encoders: reconstruction versus compression
We discuss the similarities and differences between training an auto-encoder
to minimize the reconstruction error, and training the same auto-encoder to
compress the data via a generative model. Minimizing a codelength for the data
using an auto-encoder is equivalent to minimizing the reconstruction error plus
some correcting terms which have an interpretation as either a denoising or
contractive property of the decoding function. These terms are related but not
identical to those used in denoising or contractive auto-encoders [Vincent et
al. 2010, Rifai et al. 2011]. In particular, the codelength viewpoint fully
determines an optimal noise level for the denoising criterion
Online Natural Gradient as a Kalman Filter
We cast Amari's natural gradient in statistical learning as a specific case
of Kalman filtering. Namely, applying an extended Kalman filter to estimate a
fixed unknown parameter of a probabilistic model from a series of observations,
is rigorously equivalent to estimating this parameter via an online stochastic
natural gradient descent on the log-likelihood of the observations.
In the i.i.d. case, this relation is a consequence of the "information
filter" phrasing of the extended Kalman filter. In the recurrent (state space,
non-i.i.d.) case, we prove that the joint Kalman filter over states and
parameters is a natural gradient on top of real-time recurrent learning (RTRL),
a classical algorithm to train recurrent models.
This exact algebraic correspondence provides relevant interpretations for
natural gradient hyperparameters such as learning rates or initialization and
regularization of the Fisher information matrix.Comment: 3rd version: expanded intr
An inverse Satake isomorphism in characteristic p
Let F be a local field with finite residue field of characteristic p and k an
algebraic closure of the residue field. Let G be the group of F-points of a
F-split connected reductive group. In the apartment corresponding to a chosen
maximal split torus of T, we fix a hyperspecial vertex and denote by K the
corresponding maximal compact subgroup of G. Given an irreducible smooth
k-representation of K, we construct an isomorphism from the affine
semigroup k-algebra of the dominant cocharacters of T onto the Hecke algebra
. In the case when the derived subgroup of G is simply connected,
we prove furthermore that our isomorphism is the inverse to the Satake
isomorphism constructed by Herzig
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