932 research outputs found
Recovery Guarantees for Quadratic Tensors with Limited Observations
We consider the tensor completion problem of predicting the missing entries
of a tensor. The commonly used CP model has a triple product form, but an
alternate family of quadratic models which are the sum of pairwise products
instead of a triple product have emerged from applications such as
recommendation systems. Non-convex methods are the method of choice for
learning quadratic models, and this work examines their sample complexity and
error guarantee. Our main result is that with the number of samples being only
linear in the dimension, all local minima of the mean squared error objective
are global minima and recover the original tensor accurately. The techniques
lead to simple proofs showing that convex relaxation can recover quadratic
tensors provided with linear number of samples. We substantiate our theoretical
results with experiments on synthetic and real-world data, showing that
quadratic models have better performance than CP models in scenarios where
there are limited amount of observations available
Training Input-Output Recurrent Neural Networks through Spectral Methods
We consider the problem of training input-output recurrent neural networks
(RNN) for sequence labeling tasks. We propose a novel spectral approach for
learning the network parameters. It is based on decomposition of the
cross-moment tensor between the output and a non-linear transformation of the
input, based on score functions. We guarantee consistent learning with
polynomial sample and computational complexity under transparent conditions
such as non-degeneracy of model parameters, polynomial activations for the
neurons, and a Markovian evolution of the input sequence. We also extend our
results to Bidirectional RNN which uses both previous and future information to
output the label at each time point, and is employed in many NLP tasks such as
POS tagging
Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank- Updates
In this paper, we provide local and global convergence guarantees for
recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the
proposed algorithm is a simple alternating rank- update which is the
alternating version of the tensor power iteration adapted for asymmetric
tensors. Local convergence guarantees are established for third order tensors
of rank in dimensions, when and the tensor
components are incoherent. Thus, we can recover overcomplete tensor
decomposition. We also strengthen the results to global convergence guarantees
under stricter rank condition (for arbitrary constant ) through a simple initialization procedure where the algorithm is
initialized by top singular vectors of random tensor slices. Furthermore, the
approximate local convergence guarantees for -th order tensors are also
provided under rank condition . The guarantees also
include tight perturbation analysis given noisy tensor.Comment: We have added an additional sub-algorithm to remove the (approximate)
residual error left after the tensor power iteratio
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
Tensor decomposition with generalized lasso penalties
We present an approach for penalized tensor decomposition (PTD) that
estimates smoothly varying latent factors in multi-way data. This generalizes
existing work on sparse tensor decomposition and penalized matrix
decompositions, in a manner parallel to the generalized lasso for regression
and smoothing problems. Our approach presents many nontrivial challenges at the
intersection of modeling and computation, which are studied in detail. An
efficient coordinate-wise optimization algorithm for (PTD) is presented, and
its convergence properties are characterized. The method is applied both to
simulated data and real data on flu hospitalizations in Texas. These results
show that our penalized tensor decomposition can offer major improvements on
existing methods for analyzing multi-way data that exhibit smooth spatial or
temporal features
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