Deep learning and the renormalization group
Authors: Cédric Bény
ICLR 2013

Abstract: Renormalization group methods, which analyze the way in which the effective behavior of a system depends on the scale at which it is observed, are key to modern condensed-matter theory and particle physics. The aim of this paper is to compare and contrast the ideas behind the renormalization group (RG) on the one hand and deep machine learning on the other, where depth and scale play a similar role. In order to illustrate this connection, we review a recent numerical method based on the RG---the multiscale entanglement renormalization ansatz (MERA)---and show how it can be converted into a learning algorithm based on a generative hierarchical Bayesian network model. Under the assumption---common in physics---that the distribution to be learned is fully characterized by local correlations, this algorithm involves only explicit evaluation of probabilities, hence doing away with sampling.

Deep learning and the renormalization group
C´edric B´eny
Institut f¨ur Theoretische Physik
Leibniz Universit¨at Hannover
Appelstraße 2, 30167 Hannover, Germany
cedric.beny@gmail.com
Abstract
Renormalization group (RG) methods, which model the way in which the effec-
tive behavior of a system depends on the scale at which it is observed, are key to
modern condensed-matter theory and particle physics. We compare the ideas be-
hind the RG on the one hand and deep machine learning on the other, where depth
and scale play a similar role. In order to illustrate this connection, we review a
recent numerical method based on the RG—the multiscale entanglement renor-
malization ansatz (MERA)—and show how it can be converted into a learning
algorithm based on a generative hierarchical Bayesian network model. Under the
assumption—common in physics—that the distribution to be learned is fully char-
acterized by local correlations, this algorithm involves only explicit evaluation of
probabilities, hence doing away with sampling.
Natural sciences extend the process by which we intuitively discover patterns in our sensory data
to domains beyond our natural sensory abilities, but also beyond our intuitive reasoning abilities.
Already, machine learning methods are becoming important in ﬁelds where the patterns are too
complex to be modeled with simple equations, and too far removed from intuition to be compre-
hended without formal tools.
Physics may seem an unlikely candidate for application of machine learning, given that it speciﬁcally
focuses on systems which can be understood with simple laws. Nevertheless, these very compact
laws have to be compared to experiments, and hence require a “decompression” process which
rapidly leads to overwhelming levels of complexity.
The renormalization group (RG) was designed to handle some of these complexities and had a
tremendous inﬂuence in particle physics and condensed-matter physics [1]. It models the depen-
dence of a system’s effective behavior on a parameter which is usually thought of asscale or energy,
but which we will think of here as depth. Although the RG idea is not recent, new approaches have
emerged which have the potential to fully automatize it [2–5]. These methods are designed for sim-
ulation rather than learning, but, because they are deﬁned through clever representations of certain
classes of (quantum) states, they can be adapted to learning tasks as well [6, 7].
In this paper, I will explain how one of these methods (the multiscale entanglement renormaliza-
tion ansatz [8]) can be made into a deep learning algorithm for classical probability distributions,
assuming no prior knowledge of RG or MERA, nor of quantum mechanics. The machine learning
framework task that I am considering is that of building a representation of a probability distribution
which optimizes the likelihood of the training data.
1 The renormalization group
Let us take the point of view of condensed matter physics, where the system of interest is a d-
dimensional lattice which we think of as a graph Γ which could be embedded in Rd in such a way
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arXiv:1301.3124v4  [quant-ph]  13 Mar 2013
that the edges loosely indicate proximity of sites. Hence we can think of it as a discrete version
of a d-dimensional space. We take the graph as ﬁxed once and for all, but we associate with each
vertex v, which we call a site, a random variable Xv which can take any of npossible values. The
joint probability distribution for all these variables will be referred to as the lattice’sstate, and is the
object that we want to model.
In the context of machine learning, an example would be a probability distribution over images. In
this example, the dimension is d = 2 , the sites are pixels linked to their four neighbors, and the
random variables are the pixels color values.
Physicists are often interested in such system at thermal equilibrium, where the state can be encoded
in a ridiculously compact way through the system’s Hamiltonian, or energy function, which is pro-
portional to the logarithm of the probability distribution. What makes this description compact is
that the Hamiltonian is usually a sum of local functions, i.e., functions which depend only on a little
group of neighboring sites. Hence, the state is a Markov random ﬁeld with respect to the graph Γ.
However, deducing any property of the state from the Hamiltonian (such as marginal probabilities)
is generally very difﬁcult. In fact, the large majority of the work done in theoretical physics is purely
concerned with doing just that: deriving approximate properties of systems whose state is deﬁned
by a given Hamiltonian, i.e., decompressing the Hamiltonian description, so that predictions can be
compared to experiments. In this sense, the Hamiltonian, although it is a compact representation of
the state, is not very useful as a representation of what one knows about the system’s properties.
A Hamiltonian is not, however, entirely useless. Given a Hamiltonian, it is relatively easy to make
predictions about short-range correlations. For instance, by sampling the distribution using a Monte-
Carlo simulation. However, the computational cost grows exponentially with the range of the corre-
lations.
The RG approach is based on the observation that longer range correlations can often be summarized
by an effective Hamiltonian on a smaller array of coarse-grained random variables which are, for
instance, averages of neighboring sites. That is, one can build a stochastic map from the state of
the system to that of coarser array of new variables in such a way that the new state can also be
described by a local Hamiltonian. This process can be iterated, leading to a hierarchy of effective
Hamiltonians labeled by a scale. This family, if it could be computed, would provide an efﬁcient
encoding of the system’s behavior at all scales.
The actual choice of “binning”, i.e., the form of the stochastic maps which implement the coarse-
grainings, is an essential element of the process, as it selects the proper, emergent,order parameters
which contain non-trivial information about the large-scale properties of the system. One may think
of these local order parameters as higher-order concepts needed to understand the full state. Typi-
cally, the only variables left at the largest scale are the thermodynamic variables which are averages
over the whole system. If the coarse-graining procedure is chosen correctly, those variables should
identify in which thermodynamic phase the system ﬁnds itself.
This hierarchical scale-by-scale encoding of a probability distribution resembles, for instance, a
deep encoding of an image whose features are deﬁned by iteratively combining locale patterns into
more complex—and larger—patterns. Where this analogy fails, however, is that the distribution
to be learned in the context of image recognition is likely not generated by a local Hamiltonian.
Nevertheless, we may still think of a local Hamiltonian which appropriately reproduces the short-
range correlations within neighboring pixels. For instance, this Hamiltonian could assign higher
probabilities to edge ﬁlters. The edge orientations themselves may then be the next coarser variables
governed by their own effective Hamiltonian which prefers certain assembly of neighboring edges,
and so on until, at the deepest level, the only variables left encodes what the whole image represents,
with an effective Hamiltonian yielding the relative frequency of the different classes in the learning
data.
The renormalization “group” denotes the set of coarse-graining operations with composition as
group operation, although it is rarely a group given that its elements typically have no inverse,
and cannot all be composed together. It is more fruitful to think of it as a ﬂow on the “manifold”
of Hamiltonians (assuming a continuum of scales), where each possible Hamiltonian ﬂows toward
its effective version at larger and larger scales. This picture, however, relies on the Hamiltonian
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uniquely deﬁning the state, and must be given up if we trust the Hamiltonians to encode only the
short-range correlations.
The previous discussion is rather idealized, as a lot of work is needed to transform these ideas into
concrete methods. This is why most techniques classiﬁed under the umbrella of RG are very model-
speciﬁc and hence of little use for us.
However, the study of quantum phase transitions (i.e. phase transitions at zero temperatures, where
all ﬂuctuations are quantum instead of thermal) has motivated the appearance of much more generic
numerical methods based on an ansatz, i.e., a parameterization of a manageably small set of states
which happen to nearly contain many of the interesting physical states. For instance, the very suc-
cessful density-matrix renormalization group (DMRG) can be understood as an optimization method
within the matrix product state (MPS) ansatz. Classically, an MPS can be thought of as a type of
stochastic ﬁnite-state machine (although another classical version has been proposed [9]). This,
however, only works on one-dimensional systems, at least in the quantum setting. In addition, the
connection with the RG is somewhat weak (even though it is faithful to a proposal made in the
seminal paper by Wilson on the renormalization group [1]).
Here, we will consider instead a more recent proposal which can be formulated in any dimension,
and which more directly incorporates the idea of a multiscale description of the system.
2 Stochastic version of MERA
The multiscale entanglement renormalization ansatz (MERA) is an efﬁcient parametrization of a
certain subset of quantum states (which one may think of as probability distributions but where
the probabilities take complex values). It is numerically efﬁcient in two ways which are important
for its use in physics. Firstly, the number of parameters involved typically grows linearly with the
number N of sites. Secondly, marginal distributions over a constant number of sites (not necessarily
contiguous) can be explicitly computed in a time of order log N.
Instead of describing the full quantum MERA, I will only introduce a classical version of it. Since
all entanglement (a.k.a. quantum correlations) is replaced by classical correlations, let us call it the
correlation renormalization ansatz (CORA).
For simplicity, I will describe the model as it would apply to data living on a square d-dimensional
lattice, such as a time series for d = 1 or an image for d = 2 , and assume a renormalization
model where the coarse-graining is done by grouping blocks of 2 ×2 ×... = 2d neighboring sites.
Generalization to other lattices or other coarse-graining scheme are rather straightforward.
Instead of starting with an algorithm per se, we start with a “knowledge representation”, that is,
a parameterization of a certain set of probability distributions on the observed data. Provided that
this parameterization can be stored efﬁciently, and that the cost function for the learning process be
computable efﬁciently, then an algorithm can be devised to minimize the cost function.
We call Γ0 the original square lattice on which the data lives, and Γj, j = 1,2,... the successive
coarse-graining of that lattice. Each site of Γj must be thought of as representing a block of 2d sites
of Γj−1, in the sense that the neighborhood relations between the sites of Γj represent that of the
corresponding blocks on Γj−1.
Since the number of sites of Γj is a constant fraction of that of Γj−1 (1/2d in this case), the total
number of coarse-graining steps is logarithmic in the total size N of Γ0, i.e. in the total number of
variables.
By a state on Γj, we refer to a joint probability distribution for the values of the variables associated
with each site of Γj. We denote the set of these distributions by S(Γj). The idea is to represent a
state µ∈S(Γ0) through a sequence of stochastic maps
πj : S(Γj) →S(Γj−1)
as
µ= π1 ◦π2 ◦... ◦πjmax (e),
where ◦denotes composition, and e is some ﬁducial state on the coarsest lattice Γjmax , e.g., the
uniform distribution. In the original MERA, these mapsπj are isometric operator on a Hilbert space,
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Figure 1: Example of a MERA for a one-dimensional lattice, where only two layers are represented.
The boxes and triangles are arbitrary isometric maps between Hilbert spaces represented by the
vertical lines. We obtain a CORA by replacing the boxes and triangles by stochastic maps between
random variables. This diagram can then be read as an operational recipe to produce the state on the
lattice Γ0, from that of Γ2: each vertical line represents some data, or random variable, and a box
or triangle is a speciﬁc stochastic map which must be applied to the joint probability distribution
of all its input(s) (the lines coming in from above) in order to produce the joint distribution of its
outputs (represented by the lines coming out below it). This can be equivalently represented by the
Bayesian network of Figure 2. The parameters of the model are the components of all the local
stochastic maps (boxes and triangles), together with the initial state of the last layer Γjmax .
which is the quantum equivalent of stochastic maps, in the sense that they are the most general maps
preserving the basic properties of the states.
One may think of theseπj as the inverse of the coarse-graining channels deﬁning the renormalization
group. Indeed, instead of destroying information by mapping a precise lattice to a coarser one with
fewer sites, it introduces information by creating additional sites, hence constructing the full state
scale by scale. In this sense, MERA is a “generative” interpretation of the RG.
In the original MERA, these maps are speciﬁcally required to be implementable by local maps
(stochastic maps in the classical case or isometries in the quantum case), such as represented in
Figures 1 and 2 for a one-dimensional lattice. Such a given structure being ﬁxed, the variational
parameters left to be learned are the components of each local map.
The required decomposability of the mapsπj into local operations renders the number of variational
parameters linear in the lattice size. Apart from this, however, the particular way in which the
local maps are combined is not essential, except for the causal properties of the complete maps πj
allowed: Firstly, πj : S(Γj) →S(Γj−1) must be such that, through it, each site of Γj can only
causally inﬂuence the block of sites of Γj−1 that it represents, or their neighbors as well, up to a
constant distance. In Figure 1 for instance, it is apparent that a given site can only inﬂuence the
values of the block of two sites that it represents, immediately below it, together with their nearest
neighbors.
The concept of causality that is used is deﬁned as follows: for a given stochastic matrix with several
input and output variables, we say that the ith input cannot causally inﬂuence the jth output if the
variable i is not required for the calculation of the marginal over j, no matter what the complete
input is. That is, the marginal over j can be computed purely from the marginal over all original
input variables but the ith.
Secondly, something which is also apparent in Figure 1, is the fact that πj cannot introduce any
correlations between far-away sites if its input is uncorrelated, where “far-away” means that these
output sites cannot be inﬂuenced by the same input site, in terms of the ﬁrst causality condition.
It is in this sense that each layer πj can only create, and therefore represent, correlations at a given
scale.
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Figure 2: Example of Figure 1 represented as a Bayesian network (only two layers are represented).
The bottom nodes are observed. Note that the graph is truncated, as the nodes of Γ2 must be linked
to the next layer which is not represented, as well as to each other, in the same manner as the two
layers below it.
Figure 3: Stochastic maps involved in the last two steps of the computation of the marginal state
on 3 consecutive output sites. The lines ending abruptly indicates that the corresponding variable
is summed over. The “past” of any region of Γ0 of size Lalways involves just 3 sites before level
Γ[log2(L)].
3 Learning CORA
The causal properties inherent in the deﬁnition of MERA/CORA imply that a marginal over any
ﬁnite group of L sites can be computed (explicitly, i.e., without sampling) in a time of order
eL log(N). Indeed, due to the particular causal structure of the maps πj, the past of any set of
sites of Γj, namely those sites of Γj+1 on which their values depend explicitly through πj, always
ends up involving a constant number of sites independent of N (and generally manageably small).
This is illustrated in Figure 3.
In the quantum physical setting for which MERA was introduced, the state that we want to represent
is not deﬁned by samples, but instead by a Hamiltonian, or energy function, that it minimizes,
i.e., the cost function itself. Most often, the Hamiltonians considered are local, which implies that
the evaluation of their expectation only requires the use of marginal states over small clusters of
neighboring sites. Therefore the cost function can be evaluated efﬁciently and exactly.
Such a procedure can be adapted to a situation where, instead of being handed the Hamiltonian, we
are given samples from the unknown distribution: the training data. In physics, this situation presents
itself when an experimentalists wants to reconstruct a state that he has access to only through exper-
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iments (which yields the samples), a process which is called “state tomography”. This is, of course,
no different from a typical learning task.
A difference, however, is that in physics one is often interested in states which minimize some
unknown local Hamiltonian. This assumptions implies that it is sufﬁcient to work with the efﬁciently
computable marginals over small groups of neighboring sites (so called reduced states in quantum
theory). Therefore, we can set the parameters of the channels πj by maximizing the probabilities
that these reduced states assign to the training data. In the quantum context, this was proposed
in Ref. [7]. For CORA, the cost function can be calculated in the same way, but the optimization
procedure must be slightly modiﬁed to account for the fact that the constraints on the stochastic
maps components are different from the quantum MERA.
3.1 States not determined by local laws
It is clear, however, that a MERA, or CORA, can represent a much vaster class of states, in particular
states whose long-range correlation structure is not determined by the nature of the short range
correlations, such as one would expect for natural or artiﬁcial images for instance. Indeed, the
components of the stochastic maps πj can be modiﬁed directly at any scalejwith minimal effect on
the short range correlations.
Let us therefore consider learning in the more general situation. Mirroring our discussion of sec-
tion 1, we can still use the efﬁciently computable local reduced states in order to optimize the
component of the ﬁrst channel π1, since it cannot anyway encode information about longer ranged
correlations. We say that we are working with training length ϵ0, ignoring all correlations in the
data at larger lengthscales.
This can be done in two different ways. We may simply optimize π1 by locally comparing the
training data to π1(e) where eis the uniform distribution, or any other prior (method 1), or we can
already optimize all layers simultaneously as would be done for a tomography problem, focusing
only on short-range correlation (method 2). This would have the advantage of already making long
range—or deep—deductions from the data. For instance, if all neighboring pixels were the same
color, this would already deduce that the whole image is uniform.
In a second pass, we need to compare the data to marginals generated over the new training length
ϵ1 associated with the next channel π2. We do not want to do this by comparing the data to the full
candidate state generated at the level of Γ0, as the number of sites involved would grow by a factor
2d with a bad exponential increase in complexity. Instead, we want to directly compare the reduced
state at level Γ1 to a coarse-grained (i.e., “renormalized”) version of the training data at that level.
Performing this step is where the difﬁculties lie, and could likely not be done quantum mechanically.
Classically, however, it may be possible to use sampling in this step. This coarse-graining must be
done by the Bayesian inverse of π1, with prior e(method 1), or a prior generated by the previous
layers (method 2), which is possible because we really only need the prior over scaleϵ1. An example
of the type of Bayesian networks involved is represented in Figure 2.
This process can be iterated until all O(log N) layers/channels have been trained, and the training
length covers the whole system. Further training passes can then be done in the same way in order
to exploit the better priors generated by the already trained layers.
Acknowledgments
Helpful discussions with Tobias Osborne are gratefully acknowledged. This work was supported by
the cluster of excellence EXC 201 “Quantum Engineering and Space-Time Research”.
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Peer review:
[Review]:
It is noted that the connection between RG and multi-scale modeling has been pointed out by Candes in 

E. J. Candès, P. Charlton and H. Helgason. Detecting highly oscillatory signals by chirplet path pursuit. Appl. Comput. Harmon. Anal. 24 14-40.

where it was noted that the multi-scale basis suggested in this convex optimization approach is equivalent to the Wilson basis from his original work on RG theory in the 1970s