The Recurrent Temporal Restricted Boltzmann
Machine
Ilya Sutskever, Geoffrey Hinton, and Graham Taylor
University of Toronto
{ilya, hinton, gwtaylor}@cs.utoronto.ca
Abstract
The Temporal Restricted Boltzmann Machine (TRBM) is a probabilistic model for
sequences that is able to successfully model (i.e., generate nice-looking samples
of) several very high dimensional sequences, such as motion capture data and the
pixels of low resolution videos of balls bouncing in a box. The major disadvantage
of the TRBM is that exact inference is extremely hard, since even computing
a Gibbs update for a single variable of the posterior is exponentially expensive.
This difficulty has necessitated the use of a heuristic inference procedure, that
nonetheless was accurate enough for successful learning. In this paper we introduce
the Recurrent TRBM, which is a very slight modification of the TRBM for
which exact inference is very easy and exact gradient learning is almost tractable.
We demonstrate that the RTRBM is better than an analogous TRBM at generating
motion capture and videos of bouncing balls.
1 Introduction
Modeling sequences is an important problem since there is a vast amount of natural data, such as
speech and videos, that is inherently sequential. A good model for these data sources could be useful
for finding an abstract representation that is helpful for solving “natural” discrimination tasks (see
[4] for an example of this approach for the non-sequential case). In addition, it could be also used
for predicting the future of a sequence from its past, be used as a prior for denoising tasks, and be
used for other applications such as tracking objects in video. The Temporal Restricted Boltzmann
Machine [14, 13] is a recently introduced probabilistic model that has the ability to accurately model
complex probability distributions over high-dimensional sequences. It was shown to be able to
generate realistic motion capture data [14], and low resolution videos of 2 balls bouncing in a box
[13], as well as complete and denoise such sequences.
As a probabilistic model, the TRBM is a directed graphical model consisting of a sequence of Restricted
Boltzmann Machines (RBMs) [3], where the state of one or more previous RBMs determines
the biases of the RBM in next timestep. This probabilistic formulation straightforwardly implies a
learning procedure where approximate inference is followed by learning. The learning consists of
learning a conditional RBM at each timestep, which is easily done with Contrastive Divergence
(CD) [3]. Exact inference in TRBMs, on the other hand, is highly non-trivial, since computing even
a single Gibbs update requires computing the ratio of two RBM partition functions. The approximate
inference procedure used in [13] was heuristic and was not even derived from a variational
principle.
In this paper we introduce the Recurrent TRBM (RTRBM), which is a model that is very similar
to the TRBM, and just as expressive. Despite the similarity, exact inference is very easy in the
RTRBM and computing the gradient of the log likelihood is feasible (up to the error introduced
by the use of Contrastive Divergence). We demonstrate that the RTRBM is able to generate more
realistic samples than an equivalent TRBM for the motion capture data and for the pixels of videosof bouncing balls. The RTRBM’s performance is better than the TRBM mainly because it learns to
convey more information through its hidden-to-hidden connections.
2 Restricted Boltzmann Machines
The building block of the TRBM and the RTRBM is the Restricted Boltzmann Machine [3]. An
RBM defines a probability distribution over pairs of vectors, V ∈ {0,1} NV and H ∈ {0,1} NH (a
shorthand for visible and hidden) by the equation
P(v,h) = P(V = v,H = h) = exp(v ⊤ bV + h ⊤ bH + v ⊤ Wh)/Z (1)
where bV is a vector of biases for the visible vectors, bH is a vector of biases for the hidden vectors,
and W is the matrix of connection weights. The quantity Z = Z(bV ,bH,W) is the value of the
partition function that ensures that Eq. 1 is a valid probability distribution. The RBM’s definition
implies that the conditional distributions P(H|v) and P(V |h) are factorial (i.e., all the components
of H in P(H|v) are independent) and are given by P(H (j) = 1|v) = s(bH + W ⊤ v) (j) and
P(V (i) = 1|h) = s(bV + Wh) (i) , where s(x) (j) = (1 + exp(−x (j) )) −1 is the logistic function
and x (j) is the jth component of the vector x. In general, we use i to index visible vectors V and j
to index hidden vectors H. 1 The RBM can be slightly modified to allow the vector V to take real
values; one way of achieving this is by the definition
P(v,h) = exp(−‖v‖ 2 /2 + v ⊤ bV + h ⊤ bH + v ⊤ Wh)/Z. (2)
Using this equation does not change the form of the gradients and the conditional distribution
P(H|v). The only change it introduces is in the conditional distribution P(V |h), which is equal
to a multivariate Gaussian with parameters N(bV + Wh,I). See [18, 14] for more details and
generalizations.
The gradient of the average log probability given a dataset S, L = 1/|S| ∑
log P(v), has the
following simple form:
∂L/∂W = 〈 V · H ⊤〉
P(H|V ) ˜ P(V ) − 〈 V · H ⊤〉
(3)
P(H,V )
where ˜ P(V ) = 1/|S| ∑
v∈S δv(V ) (here δx(X) is a distribution over real-valued vectors that is
concentrated at x), and 〈f(X)〉
P(X)
is the expectation of f(X) under the distribution P . Computing
the exact values of the expectations 〈·〉
P(H,V )
is computationally intractable, and much work has
been done on methods for computing approximate values for the expectations that are good enough
for practical learning and inference tasks (e.g., [16, 12, 19], including [15], which works well for
the RBM).
We will approximate the gradients with respect to the RBM’s parameters using the Contrastive
Divergence [3] learning procedure, CDn, whose updates are computed by the following algorithm.
Algorithm 1 (CDn)
1. Sample (v,h) ∼ P(H|V ) ˜ P(V )
2. Set ∆W to v · h ⊤
3. repeat n times: sample v ∼ P(V |h), then sample h ∼ P(H|v)
4. Decrease ∆W by v · h ⊤
Models learned by CD1 are often reasonable generative models of the data [3], but if learning is
continued with CD25, the resulting generative models are much better [11]. The RBM also plays a
critical role in deep belief networks [4], [5], but we do not use this connection in this paper.
3 The TRBM
It is easy to construct the TRBM with RBMs. The TRBM, as described in the introduction, is
a sequence of RBMs arranged in such a way that in any given timestep, the RBM’s biases depend
only on the state of the RBM in the previous timestep. In its simplest form, the TRBM can
1
We use uppercase variables (as in P(H|v)) to denote distributions and lowercase variables (as in P(h|v))
to denote the (real-valued) probability P(H = h|v).
v∈SFigure 1: The graphical structure of a TRBM: a directed sequence of RBMs.
be viewed as a Hidden Markov Model (HMM) [9] with an exponentially large state space that
has an extremely compact parameterization of the transition and the emission probabilities. Let
X tB
tA
= (XtA,...,XtB ) denote a sequence of variables. The TRBM defines a probability distribution
P(V T
1 = v T 1 ,H T 1 = h T 1 ) by the equation
P(v T 1 ,h T 1 ) =
T∏
P(vt,ht|ht−1)P0(v1,h1) (4)
t=2
which is identical to the defining equation of the HMM. The conditional distribution P(Vt,Ht|ht−1)
is that of an RBM, whose biases for Ht are a function of ht−1. Specifically,
P(vt,ht|ht−1) = exp ( v ⊤ t bV + v ⊤ t Wht + h ⊤ t (bH + W ′ ht−1) ) /Z(ht−1) (5)
where bV , bH and W are as in Eq. 1, while W ′ is the weight matrix of the connections from Ht−1
to Ht, making bH + W ′ ht−1 be the bias of RBM at time t. In this equation, V ∈ {0,1} NV
and
H ∈ {0,1} NH ; it is easy to modify this definition to allow V to take real values as was done in Eq. 2.
The RBM’s partition function depends on ht−1, because the parameters (i.e., the biases) of the RBM
at time t depend on the value of the random variable Ht−1. Finally, the distribution P0 is defined
by an equation very similar to Eq. 5, except that the (undefined) term W ′ h0 is replaced by the
term binit, so the hidden units receive a special initial bias at P0; we will often write P(V1,H1|h0)
for P0(V1,H1) and W ′ h0 for binit. It follows from these equations that the TRBM is a directed
graphical model that has an (undirected) RBM at each timestep (a related directed sequence of
Boltzmann Machines has been considered in [7]).
As in most probabilistic models, the weight update is computed by solving the inference problem
and computing the weight update as if the inferred variables were observed. fully-visible case. If
the hidden variables are observed, equation 4 implies that the gradient of the log likelihood with
respect to the TRBM’s parameters is ∑T
t=1 ∇log P(vt,ht|ht−1), and each term, being the gradient
of the log likelihood of an RBM, can be approximated using CDn. Thus the main computational
difficulty of learning TRBMs is in obtaining samples from a distribution approximating the posterior
P(H T 1 |vT 1 ).
Inference in a TRBM
Unfortunately, the TRBM’s inference problem is harder than that of a typical undirected graphical
model, because even computing the probability P(H (j)
t = 1| everything else) involves evaluating
the exact ratio of two RBM partition functions, which can be seen from Eq. 5. This difficulty necessitated
the use of a heuristic inference procedure [13], which is based on the observation that the
distribution P(Ht|h t−1
1 ,v t 1) = P(Ht|ht−1,vt) is factorial by definition. This inference procedure
does not do any kind of smoothing from the future and only does approximate filtering from the past
by sampling from the distribution ∏T
t−1
P(Ht|H1 ,v t 1) instead of the true posterior distribution
∏ T
t=1
t=1
t−1
P(Ht|H1 ,v T 1 ), which is easy because P(Ht|h t−1
1 ,v t 1) is factorial. 2
4 Recurrent TRBMs
Let us start with notation. Consider an arbitrary factorial distribution P ′ (H). The statement h ∼
P ′ (H) means that h is sampled from the factorial distribution P ′ (H), so each h (j) is set to 1 with
2
This is a slightly simplified description of the inference procedure in [13].Figure 2: The graphical structure of the RTRBM, Q. The variables Ht are real valued while the
variables H ′ t are binary.
The conditional distribution Q(Vt,H ′ t|ht−1) is given by the equation
Q(vt,h ′ t|ht−1) = exp ( v ⊤ t Wh ′ t + v ⊤ t bV + h ′ t(bH + W ′ ht−1) ) /Z(ht−1), which is essentially the
same as the TRBM’s conditional distribution P from equation 5. We will always integrate out H ′ t
and will work directly with the distribution Q(Vt|ht−1). Notice that when V1 is observed, H ′ 1 cannot
affect H1.
probability P ′ (H (j) = 1), and 0 otherwise. In contrast, the statement h ← P ′ (H) means that each
h (j) is set to the real value P ′ (H (j) = 1), so this is a “mean-field” update [8, 17]. The symbol P
stands for the distribution of some TRBM, while the symbol Q stands for the distribution defined by
an RTRBM. Note that the outcome of the operation · ← P(Ht|vt,ht−1) is s(Wvt +W ′ ht−1 +bH).
An RTRBM, Q(V T
1 ,H T 1 ), is defined by the equation
Q(v T 1 ,h T 1 ) =
T∏
Q(vt|ht−1)Q(ht|vt,ht−1) · Q0(v1). Q0(h1|v1) (6)
t=2
The terms appearing in this equation will be defined shortly.
Let us contrast the generative process of the two models. To sample from a TRBM P , we need
to perform a directed pass, sampling from each RBM on every timestep. One way of doing this is
described by the following algorithm.
Algorithm 2 (for sampling from the TRBM):
for 1 ≤ t ≤ T :
1. sample vt ∼ P(Vt|ht−1)
2. sample ht ∼ P(Ht|vt,ht−1) 3
where step 1 requires sampling from the marginals of a Boltzmann Machine (by integrating out Ht),
which involves running a Markov chain.
By definition, RTRBMs and TRBMs are parameterized in the same way, so from now on we will
assume that P and Q have identical parameters, which are W,W ′ ,bV ,bH, and binit. The following
algorithm samples from the RTRBM Q under this assumption.
Algorithm 3 (for sampling from the RTRBM)
for 1 ≤ t ≤ T :
1. sample vt ∼ P(Vt|ht−1)
2. set ht ← P(Ht|vt,ht−1)
We can infer that Q(Vt|ht−1) = P(Vt|ht−1) because of step 1 in Algorithm 3, which is also consistent
with the equation given in figure 2 where H ′ t is integrated out. The only difference between
Algorithm 2 and Algorithm 3 is in step 2. The difference may seem small, since the operations
ht ∼ P(Ht|vt,ht−1) and ht ← P(Ht|vt,ht−1) appear similar. However, this difference significantly
alters the inference and learning procedures of the RTRBM; in particular, it can already be
seen that Ht are real-valued for the RTRBM.
3
When t = 1, P(Ht|vt, ht−1) stands for P0(H1|v1), and similarly for other conditional distributions. The
same convention is used in all algorithms.4.1 Inference in RTRBMs
Inference in RTRBMs given v T 1 is very easy, which might be surprising in light of its similarity to
the TRBM. The reason inference is easy is similar to the reason inference in square ICAs is easy [1]:
There is a unique and an easily computable value of the hidden variables that has a nonzero posterior
probability. Suppose, for example, that the value of V1 is v1, which means that v1 was produced at
the end of step 1 in Algorithm 3. Since step 2, the deterministic operation h1 ← P0(H1|v1), has
been executed, the only value h1 can take is the value assigned by the operation · ← P0(H1|v1). Any
other value for h1 is never produced by a generative process that outputs v1 and thus has posterior
probability 0. In addition, by executing this operation, we can recover h1. Thus, Q0(H1|v1) =
δ s(Wv1+bH+binit)(H1). Note that H1’s value is completely independent of v T 2 .
Once h1 is known, we can consider the generative process that produced v2. As before, since v2
was produced at the end of step 1, then the fact that step 2 has been executed implies that h2 can be
computed by h2 ← P(H2|v2,h1) (recall that at this point h1 is known with absolute certainty). If
the same reasoning is repeated t times, then all of ht 1 is uniquely determined and is easily computed
when V t
1 is known. There is no need for smoothing because Vt and Ht−1 influence Ht with such
strength that the knowledge of V T
t+1 cannot alter the model’s belief about Ht. This is because
Q(Ht|vt,ht−1) = δ s(Wvt+bH+W ′
ht−1)(Ht).
The resulting inference algorithm is simple:
Algorithm 4 (inference in RTRBMs)
for 1 ≤ t ≤ T :
1. ht ← P(Ht|vt,ht−1)
Let h(v) T 1 denote the output of the inference algorithm on input v T 1 , in which case the posterior is
described by
Q(H T 1 |v T 1 ) = δ h(v) T
1
(H T 1 ). (7)
4.2 Learning in RTRBMs
Learning in RTRBMs may seem easy once inference is solved, since the main difficulty in learning
TRBMs is the inference problem. However, the RTRBM does not allow EM-like learning because
the equation ∇log Q(vT 1 ) = 〈 ∇log Q(vT 1 ,hT 1 ) 〉
hT 1 ∼Q(HT 1 |vT is not meaningful. To be precise,
1 )
the gradient ∇log Q(vT 1 ,hT 1 ) is undefined because δs(W ′ ht−1+bH+W T vt)(ht) is not, in general, a
continuous function of W . Thus, the gradient has to be computed differently.
Notice that the RTRBM’s log probability satisfies log Q(vT 1 ) = ∑T
t−1
t=1
log Q(vt|v1 ), so we could
try computing the sum ∇ ∑T
t−1
log Q(vt|v1 ). The key observation that makes the computation
feasible is the equation
t=1
Q(Vt|v t−1
1 ) = Q(Vt|h(v)t−1) (8)
where h(v)t−1 is the value computed by the RTRBM inference algorithm with inputs vt 1. This equation
holds because Q(vt|v t−1
1 ) = ∫
h ′ t−1
Q(vt|h ′ t−1)Q(h ′ t−1|v t−1
1 )dh ′ t−1 = Q(vt|h(v)t−1), as the
posterior distribution Q(Ht−1|v t−1
1 ) = δ h(v)t−1 (Ht−1) is a point-mass at h(v)t−1, which follows
from Eq. 7.
The equality Q(Vt|v t−1
1 ) = Q(Vt|h(v)t−1) allows us to define a recurrent neural network (RNN)
[10] whose parameters are identical to those of the RTRBM, and whose cost function is equal to the
log likelihood of the RTRBM. This is useful because it is easy to compute gradients with respect to
the RNN’s parameters using the backpropagation through time algorithm [10]. The RNN has a pair
of variables at each timestep, {(vt,rt)} T t=1, where vt are the input variables and rt are the RNN’s
hidden variables (all of which are deterministic). The hiddens r T 1 are computed by the equation
rt = s(Wvt + bH + W ′ rt−1) (9)
where W ′ rt−1 is replaced with binit when t = 1. This definition was chosen so that the equation
r T 1 = h(v) T 1 would hold. The RNN attempts to probabilistically predict the next timestep from its
history using the marginal distribution of the RBM Q(Vt+1|rt), so its objective function at time t is
defined to be log Q(vt+1|rt), where Q depends on the RNN’s parameters in the same way it dependson the RTRBM’s parameters (the two sets of parameters being identical). This is a valid definition
of an RNN whose cumulative objective for the sequence v T 1 is
O =
T∑
log Q(vt|rt−1) (10)
t=1
where Q(v1|r0) = Q0(v1). But since rt as computed in equation 9 on input vT 1 is identical to h(v)t,
the equality log Q(vt|rt−1) = log Q(vt|v t−1
1 ) holds. Substituting this identity into Eq. 10 yields
T∑
T∑
O = log Q(vt|rt−1) =
t=1
t=1
which is the log probability of the corresponding RTRBM.
log Q(vt|v t−1
1 ) = log Q(v T 1 ) (11)
This means that ∇O = ∇log Q(v T 1 ) can be computed with the backpropagation through time algorithm
[10], where the contribution of the gradient from each timestep is computed with Contrastive
Divergence.
4.3 Details of the backpropagation through time algorithm
The backpropagation through time algorithm is identical to the usual backpropagation algorithm
where the feedforward neural network is turned “on its side”. Specifically, the algorithm maintains
a term ∂O/∂rt which is computed from ∂O/∂rt+1 and ∂ log Q(vt+1|rt)/∂rt using the chain rule,
by the equation
∂O/∂rt = W ′⊤
(rt+1.(1 − rt+1).∂O/∂rt+1) + W ′⊤ ∂ log Q(vt|rt−1)/∂bH (12)
where a.b denotes component-wise multiplication, the term rt.(1 − rt) arises from the derivative of
the logistic function s ′(x) = s(x).(1 − s(x)), and ∂ log Q(vt+1|rt)/∂bH is computed by CD. Once
∂O/∂rt is computed for all t, the gradients of the parameters can be computed using the following
equations
∂O
=
∂W
′
∂O
∂W =
T∑
rt−1(rt.(1 − rt).∂O/∂rt) ⊤
t=2
T∑ −1
t=1
vt
(
W ′⊤
) ⊤
(rt+1.(1 − rt+1).∂O/∂rt+1) +
(13)
T∑
∂ log Q(vt|rt−1)/∂W (14)
The first summation in Eq. 14 arises from the use of W as weights for inference for computing rt and
the second summation arises from the use of W as RBM parameters for computing log Q(vt|rt−1).
Each term of the form ∂ log Q(vt+1|rt)/∂W is also computed with CD. Computing ∂O/∂rt is done
most conveniently with a single backward pass through the sequence. As always, log Q(v1|r0) =
Q0(v1). It is also seen that the gradient would be computed exactly if CD were to return the exact
gradient of the RBM’s log probability.
5 Experiments
We report the results of experiments comparing an RTRBM to a TRBM. The results in [14, 13] were
obtained using TRBMs that had several delay-taps, which means that each hidden unit could directly
observe several previous timesteps. To demonstrate that the RTRBM learns to use the hidden units to
store information, we did not use delay-taps for the RTRBM nor the TRBM, which causes the results
to be worse (but not much) than in [14, 13]. If delay-taps are allowed, then the results of [14, 13]
show that there is little benefit from the hidden-to-hidden connections (which are W ′ ), making the
comparison between the RTRBM and the TRBM uninteresting.
In all experiments, the RTRBM and the TRBM had the same number of hidden units, their parameters
were initialized in the same manner, and they were trained for the same number of weight
updates. When sampling from the TRBM, we would use the sampling procedure of the RTRBM
using the TRBM’s parameters to eliminate the additional noise from its hidden units. If this is not
done, the samples produced by the TRBM are significantly worse. Unfortunately, the evaluation
metric is entirely qualitative since computing the log probability on a test set is infeasible for both
the TRBM and the RTRBM. We provide the code for our experiments in [URL].
t=1Figure 3: This figure shows the receptive fields of the first 36 hidden units of the RTRBM on the
left, and the corresponding hidden-to-hidden weights between these units on the right: the ith row on
the right corresponds to the ith receptive field on the left, when counted left-to-right. Hidden units
18 and 19 exhibit unusually strong hidden-to-hidden connections; they are also the ones with the
weakest visible-hidden connections, which effectively makes them belong to another hidden layer.
5.1 Videos of bouncing balls
We used a dataset consisting of videos of 3 balls bouncing in a box. The videos are of length 100
and of resolution 30×30. Each training example is synthetically generated, so no training sequence
is seen twice by the model which means that overfitting is highly unlikely. The task is to learn to
generate videos at the pixel level. This problem is high-dimensional, having 900 dimensions per
frame, and the RTRBM and the TRBM are given no prior knowledge about the nature of the task
(e.g., by convolutional weight matrices).
Both the RTRBM and the TRBM had 400 hidden units. Samples from these models are provided as
videos 1,2 (RTRBM) and videos 3,4 (TRBM). A sample training sequence is given in video 5. All
the samples can be found in [URL]. The real-values in the videos are the conditional probabilities
of the pixels [13]. The RTRBM’s samples are noticeably better than the TRBM’s samples; a key
difference between these samples is that the balls produced by the TRBM moved in a random walk,
while those produced by the RTRBM moved in a more persistent direction. An examination of the
visible to hidden connection weights of the RTRBM reveals a number of hidden units that are not
connected to visible units. These units have the most active hidden to hidden connections, which
must be used to propagate information through time. In particular, these units are the only units that
do not have a strong self connection (i.e., W ′ i,i is not large; see figure 3). No such separation of units
is found in the TRBM and all its hidden units have large visible to hidden connections.
5.2 Motion capture data
We used a dataset that represents human motion capture data by sequences of joint angle, translations,
and rotations of the base of the spine [14]. The total number of frames in the dataset set was
3000, from which the model learned on subsequences of length 50. Each frame has 49 dimensions,
and both models have 200 hidden units. The data is real-valued, so the TRBM and the RTRBM
were adapted to have Gaussian visible variables using equation 2. The samples produced by the
RTRBM exhibit less sticking and foot-skate than those produced by the TRBM; samples from these
models are provided as videos 6,7 (RTRBM) and videos 8,9 (TRBM); video 10 is a sample training
sequence. Part of the Gaussian noise was removed in a manner described in [14] in both models.
5.3 Details of the learning procedures
Each problem was trained for 100,000 weight updates, with a momentum of 0.9, where the gradient
was normalized by the length of the sequence for each gradient computation. The weights are
updated after computing the gradient on a single sequence. The learning starts with CD10 for the
first 1000 weight updates, which is then switched to CD25. The visible to hidden weights, W , were
initialized with static CD5 (without using the (R)TRBM learning rules) on 30 sequences (which
resulted in 30 weight updates) with learning rate of 0.01 and momentum 0.9. These weights were
then given to the (R)TRBM learning procedure, where the learning rate was linearly reduced towards
0. The weights W ′ and the biases were initialized with a sample from spherical Gaussian of
standard-deviation 0.005. For the bouncing balls problem the initial learning rate was 0.01, and for
the motion capture data it was 0.005.6 Conclusions
In this paper we introduced the RTRBM, which is a probabilistic model as powerful as the intractable
TRBM that has an exact inference and an almost exact learning procedure. The common disadvantage
of the RTRBM is that it is a recurrent neural network, a type of model known to have difficulties
learning to use its hidden units to their full potential [2]. However, this disadvantage is common to
many other probabilistic models, and it can be partially alleviated using techniques such as the long
short term memory RNN [6].
Acknowledgments
This research was partially supported by the Ontario Graduate Scholarship and by the Natural Council
of Research and Engineering of Canada. The mocap data used in this project was obtained
from http://people.csail.mit.edu/ehsu/work/sig05stf/. For Matlab playback
of motion and generation of videos, we have adapted portions of Neil Lawrence’s motion capture
toolbox (http://www.dcs.shef.ac.uk/∼neil/mocap/).
References
[1] A.J. Bell and T.J. Sejnowski. An Information-Maximization Approach to Blind Separation and Blind
Deconvolution. Neural Computation, 7(6):1129–1159, 1995.
[2] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult.
Neural Networks, IEEE Transactions on, 5(2):157–166, 1994.
[3] G.E. Hinton. Training Products of Experts by Minimizing Contrastive Divergence. Neural Computation,
14(8):1771–1800, 2002.
[4] G.E. Hinton, S. Osindero, and Y.W. Teh. A Fast Learning Algorithm for Deep Belief Nets. Neural
Computation, 18(7):1527–1554, 2006.
[5] G.E. Hinton and R.R. Salakhutdinov. Reducing the Dimensionality of Data with Neural Networks. Science,
313(5786):504–507, 2006.
[6] S. Hochreiter and J. Schmidhuber. Long Short-Term Memory. Neural Computation, 9(8):1735–1780,
1997.
[7] S. Osindero and G. Hinton. Modeling image patches with a directed hierarchy of Markov random fields.
Advances Neural Information Processing Systems, 2008.
[8] C. Peterson and J.R. Anderson. A mean field theory learning algorithm for neural networks. Complex
Systems, 1(5):995–1019, 1987.
[9] L.R. Rabiner. A tutorial on hidden Markov models and selected applications inspeech recognition. Proceedings
of the IEEE, 77(2):257–286, 1989.
[10] D.E. Rumelhart, G.E. Hinton, and R.J. Williams. Learning representations by back-propagating errors.
Nature, 323(6088):533–536, 1986.
[11] R. Salakhutdinov and I. Murray. On the quantitative analysis of deep belief networks. In Proceedings of
the International Conference on Machine Learning, volume 25, 2008.
[12] D. Sontag and T. Jaakkola. New Outer Bounds on the Marginal Polytope. Advances in Neural Information
Processing Systems, 2008.
[13] I. Sutskever and G.E. Hinton. Learning multilevel distributed representations for high-dimensional sequences.
Proceeding of the Eleventh International Conference on Artificial Intelligence and Statistics,
pages 544–551, 2007.
[14] G.W. Taylor, G.E. Hinton, and S. Roweis. Modeling human motion using binary latent variables. Advances
in Neural Information Processing Systems, 19:1345–1352, 2007.
[15] T. Tieleman. Training restricted boltzmann machines using approximations to the likelihood gradient. In
Proceedings of the International Conference on Machine Learning, volume 25, 2008.
[16] M.J. Wainwright, T.S. Jaakkola, and A.S. Willsky. A new class of upper bounds on the log partition
function. IEEE Transactions on Information Theory, 51(7):2313–2335, 2005.
[17] M.J. Wainwright and M.I. Jordan. Graphical models, exponential families, and variational inference. UC
Berkeley, Dept. of Statistics, Technical Report, 649, 2003.
[18] M. Welling, M. Rosen-Zvi, and G. Hinton. Exponential family harmoniums with an application to information
retrieval. Advances in Neural Information Processing Systems, 17:1481–1488, 2005.
[19] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations.
Exploring Artificial Intelligence in the New Millennium, pages 239–236, 2003.