3,929 research outputs found
Learning hard quantum distributions with variational autoencoders
Studying general quantum many-body systems is one of the major challenges in
modern physics because it requires an amount of computational resources that
scales exponentially with the size of the system.Simulating the evolution of a
state, or even storing its description, rapidly becomes intractable for exact
classical algorithms. Recently, machine learning techniques, in the form of
restricted Boltzmann machines, have been proposed as a way to efficiently
represent certain quantum states with applications in state tomography and
ground state estimation. Here, we introduce a new representation of states
based on variational autoencoders. Variational autoencoders are a type of
generative model in the form of a neural network. We probe the power of this
representation by encoding probability distributions associated with states
from different classes. Our simulations show that deep networks give a better
representation for states that are hard to sample from, while providing no
benefit for random states. This suggests that the probability distributions
associated to hard quantum states might have a compositional structure that can
be exploited by layered neural networks. Specifically, we consider the
learnability of a class of quantum states introduced by Fefferman and Umans.
Such states are provably hard to sample for classical computers, but not for
quantum ones, under plausible computational complexity assumptions. The good
level of compression achieved for hard states suggests these methods can be
suitable for characterising states of the size expected in first generation
quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect
to v1 (rewriting of section "hard and easy quantum states", extended
discussion on comparison with tensor networks
Comparing Sample-wise Learnability Across Deep Neural Network Models
Estimating the relative importance of each sample in a training set has
important practical and theoretical value, such as in importance sampling or
curriculum learning. This kind of focus on individual samples invokes the
concept of sample-wise learnability: How easy is it to correctly learn each
sample (cf. PAC learnability)? In this paper, we approach the sample-wise
learnability problem within a deep learning context. We propose a measure of
the learnability of a sample with a given deep neural network (DNN) model. The
basic idea is to train the given model on the training set, and for each
sample, aggregate the hits and misses over the entire training epochs. Our
experiments show that the sample-wise learnability measure collected this way
is highly linearly correlated across different DNN models (ResNet-20, VGG-16,
and MobileNet), suggesting that such a measure can provide deep general
insights on the data's properties. We expect our method to help develop better
curricula for training, and help us better understand the data itself.Comment: Accepted to AAAI 2019 Student Abstrac
A New Look at an Old Problem: A Universal Learning Approach to Linear Regression
Linear regression is a classical paradigm in statistics. A new look at it is
provided via the lens of universal learning. In applying universal learning to
linear regression the hypotheses class represents the label as
a linear combination of the feature vector where ,
within a Gaussian error. The Predictive Normalized Maximum Likelihood (pNML)
solution for universal learning of individual data can be expressed
analytically in this case, as well as its associated learnability measure.
Interestingly, the situation where the number of parameters may even be
larger than the number of training samples can be examined. As expected, in
this case learnability cannot be attained in every situation; nevertheless, if
the test vector resides mostly in a subspace spanned by the eigenvectors
associated with the large eigenvalues of the empirical correlation matrix of
the training data, linear regression can generalize despite the fact that it
uses an ``over-parametrized'' model. We demonstrate the results with a
simulation of fitting a polynomial to data with a possibly large polynomial
degree
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