13,286 research outputs found
Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks
The symmetry and geometry of input data are considered to be encoded in the
internal data representation inside the neural network, but the specific
encoding rule has been less investigated. In this study, we present a
systematic method to induce a generalized neural network and its right inverse
operator, called the ridgelet transform, from a joint group invariant function
on the data-parameter domain. Since the ridgelet transform is an inverse, (1)
it can describe the arrangement of parameters for the network to represent a
target function, which is understood as the encoding rule, and (2) it implies
the universality of the network. Based on the group representation theory, we
present a new simple proof of the universality by using Schur's lemma in a
unified manner covering a wide class of networks, for example, the original
ridgelet transform, formal deep networks, and the dual voice transform. Since
traditional universality theorems were demonstrated based on functional
analysis, this study sheds light on the group theoretic aspect of the
approximation theory, connecting geometric deep learning to abstract harmonic
analysis.Comment: NeurReps 202
Probabilistic Invariant Learning with Randomized Linear Classifiers
Designing models that are both expressive and preserve known invariances of
tasks is an increasingly hard problem. Existing solutions tradeoff invariance
for computational or memory resources. In this work, we show how to leverage
randomness and design models that are both expressive and invariant but use
less resources. Inspired by randomized algorithms, our key insight is that
accepting probabilistic notions of universal approximation and invariance can
reduce our resource requirements. More specifically, we propose a class of
binary classification models called Randomized Linear Classifiers (RLCs). We
give parameter and sample size conditions in which RLCs can, with high
probability, approximate any (smooth) function while preserving invariance to
compact group transformations. Leveraging this result, we design three RLCs
that are provably probabilistic invariant for classification tasks over sets,
graphs, and spherical data. We show how these models can achieve probabilistic
invariance and universality using less resources than (deterministic) neural
networks and their invariant counterparts. Finally, we empirically demonstrate
the benefits of this new class of models on invariant tasks where deterministic
invariant neural networks are known to struggle
Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems
A new class of non-homogeneous state-affine systems is introduced for use in
reservoir computing. Sufficient conditions are identified that guarantee first,
that the associated reservoir computers with linear readouts are causal,
time-invariant, and satisfy the fading memory property and second, that a
subset of this class is universal in the category of fading memory filters with
stochastic almost surely uniformly bounded inputs. This means that any
discrete-time filter that satisfies the fading memory property with random
inputs of that type can be uniformly approximated by elements in the
non-homogeneous state-affine family.Comment: 41 page
Observing scale-invariance in non-critical dynamical systems
Recent observation for scale invariant neural avalanches in the brain have
been discussed in details in the scientific literature. We point out, that
these results do not necessarily imply that the properties of the underlying
neural dynamics are also scale invariant. The reason for this discrepancy lies
in the fact that the sampling statistics of observations and experiments is
generically biased by the size of the basins of attraction of the processes to
be studied. One has hence to precisely define what one means with statements
like `the brain is critical'.
We recapitulate the notion of criticality, as originally introduced in
statistical physics for second order phase transitions, turning then to the
discussion of critical dynamical systems. We elucidate in detail the difference
between a 'critical system', viz a system on the verge of a phase transition,
and a 'critical state', viz state with scale-invariant correlations, stressing
the fact that the notion of universality is linked to critical states.
We then discuss rigorous results for two classes of critical dynamical
systems, the Kauffman net and a vertex routing model, which both have
non-critical states. However, an external observer that samples randomly the
phase space of these two critical models, would find scale invariance. We
denote this phenomenon as 'observational criticality' and discuss its relevance
for the response properties of critical dynamical systems
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