Discriminative Recurrent Sparse Auto-Encoders
Authors: Jason Rolfe, Yann LeCun
ICLR 2013

Abstract: We present the discriminative recurrent sparse auto-encoder model, which consists of an encoder whose hidden layer is recurrent, and two linear decoders, one to reconstruct the input, and one to predict the output. The hidden layer is composed of rectified linear units (ReLU) and is subject to a sparsity penalty. The network is first trained in unsupervised mode to reconstruct the input, and subsequently trained discriminatively to also produce the desired output. The recurrent network is time-unfolded with a given number of iterations, and trained using back-propagation through time. In its time-unfolded form, the network can be seen as a very deep multi-layer network in which the weights are shared between the hidden layers. The depth allows the system to exhibit all the power of deep network while substantially reducing the number of trainable parameters. 

From an initially unstructured recurrent network, the hidden units of discriminative recurrent sparse auto-encoders naturally organize into a hierarchy of features. The systems spontaneously learns categorical-units, whose activity build up over time through interactions with part-units, which represent deformations of templates. Even using a small number of hidden units per layer, discriminative recurrent sparse auto-encoders that are pixel-permutation agnostic achieve excellent performance on MNIST.

Discriminative Recurrent Sparse Auto-Encoders
Jason Tyler Rolfe & Yann LeCun
Courant Institute of Mathematical Sciences, New York University
719 Broadway, 12th Floor
New York, NY 10003
{rolfe, yann}@cs.nyu.edu
Abstract
We present the discriminative recurrent sparse auto-encoder model, comprising a
recurrent encoder of rectiﬁed linear units, unrolled for a ﬁxed number of itera-
tions, and connected to two linear decoders that reconstruct the input and predict
its supervised classiﬁcation. Training via backpropagation-through-time initially
minimizes an unsupervised sparse reconstruction error; the loss function is then
augmented with a discriminative term on the supervised classiﬁcation. The depth
implicit in the temporally-unrolled form allows the system to exhibit far more
representational power, while keeping the number of trainable parameters ﬁxed.
From an initially unstructured network the hidden units differentiate into
categorical-units, each of which represents an input prototype with a well-deﬁned
class; and part-units representing deformations of these prototypes. The learned
organization of the recurrent encoder is hierarchical: part-units are driven di-
rectly by the input, whereas the activity of categorical-units builds up over time
through interactions with the part-units. Even using a small number of hidden
units per layer, discriminative recurrent sparse auto-encoders achieve excellent
performance on MNIST.
1 Introduction
Deep networks complement the hierarchical structure in natural data (Bengio, 2009). By breaking
complex calculations into many steps, deep networks can gradually build up complicated decision
boundaries or input transformations, facilitate the reuse of common substructure, and explicitly com-
pare alternative interpretations of ambiguous input (Lee, Ekanadham, & Ng, 2008; Zeiler, Taylor,
& Fergus, 2011). Leveraging these strengths, deep networks have facilitated signiﬁcant advances
in solving sensory problems like visual classiﬁcation and speech recognition (Dahl, et al., 2012;
Hinton, Osindero, & Teh, 2006; Hinton, et al., 2012).
Although deep networks have traditionally used independent parameters for each layer, they are
equivalent to recurrent networks in which a disjoint set of units is active on each time step. The
corresponding representations are sparse, and thus invite the incorporation of powerful techniques
from sparse coding (Glorot, Bordes, & Bengio, 2011; Lee, Ekanadham, & Ng, 2008; Olshausen &
Field, 1996, 1997; Ranzato, et al., 2006). Recurrence opens the possibility of sharing parameters
between successive layers of a deep network.
This paper introduces the Discriminative Recurrent Sparse Auto-Encodermodel (DrSAE), compris-
ing a recurrent encoder of rectiﬁed linear units (ReLU; Coates & Ng, 2011; Glorot, Bordes, &
Bengio, 2011; Jarrett, et al., 2009; Nair & Hinton, 2010; Salinas & Abbott, 1996), connected to two
linear decoders that reconstruct the input and predict its supervised classiﬁcation. The recurrent en-
coder is unrolled in time for a ﬁxed number of iterations, with the input projecting to each resulting
layer, and trained using backpropagation-through-time (Rumelhart, et al., 1986). Training initially
minimizes an unsupervised sparse reconstruction error; the loss function is then augmented with a
1
arXiv:1301.3775v4  [cs.LG]  19 Mar 2013
discriminative term on the supervised classiﬁcation. In its temporally-unrolled form, the network
can be seen as a deep network, with parameters shared between the hidden layers. The temporal
depth allows the system to exhibit far more representational power, while keeping the number of
trainable parameters ﬁxed.
Interestingly, experiments show that DrSAE does not just discover more discriminative “parts” of
the form conventionally produced by sparse coding. Rather, the hidden units spontaneously dif-
ferentiate into two types: a small number of categorical-units and a larger number of part-units.
The categorical-units have decoder bases that look like prototypes of the input classes. They are
weakly inﬂuenced by the input and activate late in the dynamics as the result of interaction with the
part-units. In contrast, the part-units are strongly inﬂuenced by the input, and encode small trans-
formations through which the prototypes of categorical-units can be reshaped into the current input.
Categorical-units compete with each other through mutual inhibition and cooperate with relevant
part-units. This can be interpreted as a representation of the data manifold in which the categorical-
units are points on the manifold, and the part-units are akin to tangent vectors along the manifold.
1.1 Prior work
The encoder architecture of DrSAE is modeled after the Iterative Shrinkage and Threshold Algo-
rithm (ISTA), a proximal method for sparse coding (Chambolle, et al., 1998; Daubechies, Defrise, &
De Mol, 2004). Gregor & LeCun (2010) showed that the sparse representations computed by ISTA
can be efﬁciently approximated by a structurally similar encoder with a less restrictive, learned pa-
rameterization. Rather than learn to approximate a precomputed optimal sparse code, the LISTA au-
toencoders of Sprechmann, Bronstein, & Sapiro (2012a,b) are trained to directly minimize the sparse
reconstruction loss function. DrSAE extends LISTA autoencoders with a non-negativity constraint,
which converts the shrink nonlinearity of LISTA into a rectiﬁed linear operator; and introduces a uni-
ﬁed classiﬁcation loss, as previously used in conjunction with traditional sparse coders (Bradley &
Bagnell, 2008; Mairal, et al., 2009; Mairal, Bach, & Ponce, 2012) and other autoencoders (Boureau,
et al., 2010; Ranzato & Szummer, 2008).
DrSAEs resemble the structure of deep sparse rectiﬁer neural networks (Glorot, Bordes, & Ben-
gio, 2011), but differ in that the parameter matrices at each layer are tied (Bengio, Boulanger-
Lewandowski, & Pascanu, 2012), the input projects to all layers, and the outputs are normalized.
DrSAEs are also reminiscent of the recurrent neural networks investigated by Bengio & Gingras
(1996), but use a different nonlinearity and a heavily regularized loss function. Finally, they are simi-
lar to the recurrent networks described by Seung (1998), but have recurrent connections amongst the
hidden units, rather than between the hidden units and the input units, and introduce classiﬁcation
and sparsiﬁcation losses.
2 Network architecture
In the following, we use lower-case bold letters to denote vectors, upper-case bold letters to denote
matrices, superscripts to indicate iterative copies of a vector, and subscripts to index the columns
(or rows, if explicitly speciﬁed by the context) of a matrix or (without boldface) the elements of a
vector. We consider discriminative recurrent sparse auto-encoders (DrSAEs) of rectiﬁed linear units
with the architecture shown in ﬁgure 1:
zt+1 = max
(
0,E ·x + S ·zt −b
)
(1)
for t= 1,...,T , where n-dimensional vector zt is the activity of the hidden units at iteration t, m-
dimensional vector x is the input, and zt=0 = 0. Unlike traditional recurrent autoencoders (Bengio,
Boulanger-Lewandowski, & Pascanu, 2012), the input projects to every iteration. We call then×m
parameter matrix E the encoding matrix, and the n×n parameter matrix S the explaining-away
matrix. The n-element parameter vector b contains a bias term. The parameters also include the
m×ndecoding matrix D and the l×nclassiﬁcation matrix C.
We pretrain DrSAEs using stochastic gradient descent on the unsupervised loss function
LU = 1
2 ·
⏐⏐⏐⏐x −D ·zT⏐⏐⏐⏐2
2 + λ·
⏐⏐⏐⏐zT⏐⏐⏐⏐
1 , (2)
2
D ̄x
C ̄y
Decoding matrix
Classification matrix
x
y
zT
L1 loss
0
z/∥z∥
L2 loss
Normalization
Logistic loss
Encoding matrix
−b
S + ()+
Ex
0 zt
Repeated T times
Explaining away Rectification
Figure 1: The discriminative recurrent sparse auto-encoder (DrSAE) architecture. zt is the hidden
representation after iterationtof T, and is initialized toz0 = 0; x is the input; andy is the supervised
classiﬁcation. Overbars denote approximations produced by the network, rather than the true input.
E, S, D, and b are learned parameters.
with the magnitude of the columns ofD bounded by 1,1 and the magnitude of the rows ofE bounded
by 1.25
T .2 We then add in the supervised classiﬁcation loss function
LS = logisticy
(
C · zT
||zT||
)
, (3)
where the multinomial logistic loss function is deﬁned by
logisticy(z) =zy −log
(∑
i
ezi
)
,
and y is the index of the desired class. 3 Starting with the parameters learned by the unsupervised
pretraining, we perform discriminative ﬁne-tune by stochastic gradient descent on LU + LS, with
the magnitude of the rows of C bounded by 5.4 The learning rate of each matrix is scaled down by
the number of times it is repeated in the network, and the learning rate of the classiﬁcation matrix
is scaled down by a factor of 5, to keep the effective learning rate consistent amongst the parameter
matrices.
We train DrSAEs with T = 11 recurrent iterations (ten nontrivial passes through the explaining-
away matrix S)5 and 400 hidden units on the MNIST dataset of28×28 grayscale handwritten digits
(LeCun, et al., 1998), with each input normalized to haveℓ2 magnitude equal to 1. We use a training
set of 50,000 elements, and a validation set of 10,000 elements to perform early-stopping. Encoding,
decoding, and classiﬁcation matrices learned via this procedure are depicted in ﬁgure 2.
The dynamics of equation 1 are inspired by the Learned Iterative Shrinkage and Thresholding Al-
gorithm (LISTA) (Gregor & LeCun, 2010), an efﬁcient approximation to the sparse coding Iterative
Shrinkage and Threshold Algorithm (ISTA) (Chambolle, et al., 1998; Daubechies, Defrise, & De
1This sets the scale ofz; otherwise, the magnitude ofz will shrink to zero and the magnitude of the columns
of D will explode. This and all other such constraints are enforced by a projection after each SGD step.
2The size of each ISTA step must be sufﬁciently small to guarantee convergence. As the step size grows
large, the input will be over-explained by multiple aligned hidden units, leading to extreme oscillations. This
bound serves the same function as ℓ2 weight regularization (Hinton, 2010). The particular value of the bound
is heuristic, and was determined by an informal search of parameter space.
3Consistent with standard autoencoders but unlike traditional applications of backpropagation-through-time,
the loss functions LU and LS only depend directly on the ﬁnal iteration of the hidden units zT .
4As in the case of the encoder, this serves the same function asℓ2 weight regularization (Hinton, 2010). The
particular value of the bound is heuristic, and was determined by an informal search of parameter space.
5The chosen number of recurrent iterations achieves a heuristic balance between representational power and
computational expense. Experiments were conducted with T ∈{2, 6, 11, 21}.
3
(a)
Enc
(b)
Dec
(c)
Clas
Figure 2: The hidden units differentiate into spatially localized part-units, which have well-aligned
encoders and decoders; and global prototype categorical-units, which have poorly aligned encoders
and decoders. A subset of the rows of encoding matrix E (a) and the columns of decoding ma-
trix D (b), and all rows of the classiﬁcation matrix C (c) after training. The ﬁrst row of (a,b) shows
the most categorical units; the last row contains the least categorical units; and the middle row evenly
steps through the remaining units in order of categoricalness. Gray pixels denote connections with
weight 0; darker pixels indicate more positive connections.
Mol, 2004). ISTA is an algorithm for minimizing theℓ1-regularized reconstruction loss function LU
of equation 2 with respect to zT. It is deﬁned by the iterative step
zt+1 = hα·λ
(
α·D⊤·x +
(
I −α·D⊤·D
)
·zt)
,
where [hθ(x)]i = sign (xi) ·max (0,|xi|−θ) and α is a small step-size parameter. With non-
negative units, ISTA is equivalent to projected gradient descent ofLU of equation 2. As the number
of iterations T →∞, a DrSAE deﬁned by equation 1 becomes a non-negative version of ISTA if it
satisﬁes the restrictions:
E = α·D⊤, S = I −α·D⊤·D , b i = α·λ, and zt
i ≥0 , (4)
where the positive scale factor αis less than the maximal eigenvalue of D⊤·D, and I is the n×n
identity matrix.
As in LISTA, but unlike ISTA, the encoding matrix E and explaining-away matrix S in a DrSAE
are independent of the decoding matrix D. Connections from the input to the hidden units, and
recurrent connections between the hidden units, are all-to-all, so the network structure is agnostic to
permutations of the input. DrSAEs can also be understood as deep, feedforward networks with the
parameter matrices tied between the layers.
3 Analysis of the hidden unit representation
Discriminative ﬁne-tuning naturally induces the hidden units of a DrSAE to differentiate into a
hierarchy-like continuum. On one extreme are part-units, which perform an ISTA-like sparse cod-
ing computation; on the other are categorical-units, which use a sophisticated form of pooling to
integrate over matching part-units, and implement winner-take-all dynamics amongst themselves.
Converging lines of evidence indicate that these two groups use distinct computational mechanisms
and serve different representational roles.
In the ISTA algorithm, each row of the encoding matrix Ei (which we sometimes call the encoder
of unit i) is proportional to the corresponding column of the decoding matrix Di (which we call the
decoder of unit i), and each row (S −I)i is proportional to (Di)⊤·D, as in equation 4. As a result,
the angle between Ei and Di, and the angle between the rows ofS −I and D⊤·D, are both simple
measures of the degree to which a unit’s dynamics follow the ISTA algorithm, and thus perform
4
11 enc iters, no discriminative training 2 enc iters, no discriminative training
(a)
 (b)
11 enc iters, with discriminative training 2 enc iters, with discriminative training
(c)
 (d)
(e)
 (f)
Figure 3: The hidden units differentiate into two populations after discriminative ﬁne-tuning. The
magnitude of row (S −I)i (a,b,e) and Ci (c,d), versus the angle between encoder row and decoder
column, for each unit from networks using 11 (a,c,e) and 2 (b,d,f) iterations. All plots are from
discriminatively ﬁne-tuned networks except (a,b), which are only subject to unsupervised pretrain-
ing. We call the dense cloud in the bottom-left part-units, and the tail extending to the top-right
categorical-units.
sparse coding.6 These quantities are equal to 0 in the case of perfect ISTA, and grow larger as the
network diverges from ISTA. Of these two angles, the explaining-away matrix comparison is more
difﬁcult to interpret, since a distortion of any one unit’s decoding column Di will affect all rows of
D⊤·D, whereas the angle between the encoder rowEi and decoder column Di only depends upon
a single unit. For this reason, we use the angle between the encoder row and decoder column as a
measure of the position of each unit on the part/categorical continuum.
6We always use S −I when plotting recurrent connection strength, since it governs the perturbation of the
otherwise stable hidden unit activations, as in projected gradient descent of LU; i.e., ISTA.
5
Figure 3 plots, for each unit i, the magnitude of row (S −I)i and column Ci, versus the angle
between row Ei and column Di. Before discriminative ﬁne-tuning, there are no categorical-units;
the angle between the encoder row and decoder column is small and the incoming recurrent con-
nections are weak for all units, as in ﬁgure 3(a,b). After discriminative ﬁne-tuning, there remains
a dense cloud of points for which the angle between the encoder row and decoder column is very
small, and the incoming recurrent and outgoing classiﬁcation connections are weak. Abutting this
is an extended tail of points that have a larger angle between the encoder row and decoder column,
and stronger incoming recurrent and outgoing classiﬁcation connections. We call units composing
the dense cloud part-units, since they have ISTA-compatible connections, while we refer to those
making up the extended tail as categorical-units, since they have strong connections to the classiﬁ-
cation output.7 When trained on MNIST, part-units have localized, pen stroke-like decoders, as can
be seen in the bottom rows of ﬁgure 2(a,b). Categorical-units, in contrast, tend to have whole-digit
prototype-like decoders, as in the top rows of ﬁgure 2(a,b). Discriminative ﬁne-tuning induces the
differentiation of categorical-units regardless of the depth of the encoder.
3.1 Part-units
Examination of the relationship between the elements of S −I and D⊤·D conﬁrms that part-
units with an encoder-decoder angle less than 0.5 radians abide by ISTA, and so perform sparse
coding on the residual input after the categorical-unit prototypes are subtracted out. The prominent
diagonals with matching slopes in ﬁgure 4(a,b), which plot the value ofSi,j−δi,j versus Di·Dj for
connections between part-units, and from categorical-units to part-units, respectively, demonstrate
that part-units receive ISTA-consistent connections from all units. The ﬁdelity of these connections
to the ISTA ideal is not strongly dependent upon whether the afferent units are ISTA-compliant part-
units, or ISTA-ignoring categorical-units. As a result, the part-units treat the categorical-units as if
they were also participating in the reconstruction of the input, and only attempt to reconstruct the
residual input not explained by the categorical-unit prototypes.
As can be seen in ﬁgure 4(c), the degree to which the encoder conforms to the ISTA algorithm
is strongly correlated with the degree to which the explaining-away matrix matches the ISTA al-
gorithm. Figure 5 shows the decoders associated with the strongest recurrent connections to three
representative part-units. As expected, the decoders of these afferent units tend to be strongly aligned
or anti-aligned with their target’s decoder, and include both part-units and categorical-units.
3.2 Categorical-units
In contrast, the recurrent connections to categorical-units with an encoder-decoder angle greater
than 0.7 radians are not strongly correlated with the values predicted by ISTA. Rather than analyz-
ing connections to the categorical-units only based upon their destination, it is more informative to
consider them organized by their source. Part-units are compatible with categorical-units of certain
classes,8 and not with others, as shown by ﬁgure 6(a). Part-units generally have positive connec-
tions to categorical-units with parallel prototypes, independent of offset, and negative connections
to categorical-units with orthogonal prototypes, as shown in ﬁgure 7(a). This corresponds to a so-
phisticated form of pooling (Jarrett, et al., 2009), with a single categorical-unit drawing excitation
from a large collection of parallel but not necessarily perfectly aligned part-units, as in ﬁgure 6(c). It
is also suggestive of the standard Hubel and Wiesel model of complex cells in primary visual cortex
(Hubel & Wiesel, 1962). ISTA would instead predict a connection proportional to the inner product,
which is zero for orthogonal prototypes and negative for anti-aligned prototypes.
Part-units use sparse coding dynamics, and so are not disproportionately suppressed by categorical-
units that represent any particular class. However, each part-unit is itself compatible with (i.e., has
positive connections to) categorical-units of only a subset of the classes. As a result, the categorical-
7For the purpose of constructing ﬁgures characterizing the difference between part-units and categorical-
units, we consider units with encoder-decoder angle less than 0.5 radians to be part-units, and units with
encoder-decoder angle greater than 0.7 radians to be categorical-units. These thresholds are heuristic, and
fail to reﬂect the continuum that exists between part- and categorical-units, but they facilitate analysis of the
extremes.
8Categorical-units have strong, sparse classiﬁcation matrix projections, as shown in ﬁgures 2(c) and 3(e,f),
and can be identiﬁed with the output class to which they have the strongest projection.
6
(a)
 (b)
(c)
Figure 4: Part-units have connections consistent with ISTA. The actual connection weights S −I
versus the ISTA-predicted weights D⊤·D, for connections from part-units to part-units (a) and
categorical-units to part-units (b); and the angle between the rows of S −I and the ISTA-ideal
D⊤·D versus the angle between the encoder rows and decoder columns (c). Units are considered
part-units if the angle between their encoder and decoder is less than 0.5 radians, and categorical-
units if the angle between their encoder and decoder is greater than 0.7 radians.
Dest Source units
Figure 5: Part-units receive ISTA-compatible connections and thus perform sparse coding on the
residual input after the contribution of the categorical-units is subtracted out. The decoders of the
twenty units with the strongest explaining-away connections |Si,j −δi,j|to three typical part-units,
sorted by connection magnitude. The left-most column depicts the decoder of the recipient part-
unit. The bars above the decoders in the remaining columns indicate the strength of the connections.
Black bars are used for positive connections, and white bars for negative connections.
units and thus the class chosen are determined by the part-unit activations. In particular, only a
subset of the possible deformations implemented by part-unit decoders are freely available for each
prototype, since part-units with a strong negative connection to a categorical-unit will tend to silence
it, and so cannot be used to transform the prototype of that categorical-unit.
Categorical-units implement winner-take-all-like dynamics amongst themselves, as shown in ﬁg-
ure 6(b), with negative connections to most other categorical-units. Positive total self-connections
Si,i facilitate the integration of inputs over time.
7
Src Destination units
(a)
(b)
Dest Source units
(c)
Figure 6: Categorical-units execute a sophisticated form of pooling over part-units, and have winner-
take-all dynamics amongst themselves. The decoders of the categorical-units receiving the twenty
strongest connections |Si,j −δi,j|from representative part-units (a) and categorical-units (b), and
the decoders of the part-units sending the twenty strongest projections to representative categorical-
units (c). The connections are sorted ﬁrst by the class of their destination, and then by the magnitude
of the connection. The left-most column depicts the decoder of the source (a,b) or destination (c)
unit. The bars above the decoders in the remaining columns indicate the strength of the connections.
Black bars are used for positive connections, and white bars for negative connections.
When activated, the categorical-units make a much larger contribution to the reconstruction than
any single part-unit, as can be seen in ﬁgure 7(b). Since, the projections from categorical-units
to part-units are consistent with ISTA, the magnitude of the categorical-unit contribution to the
reconstruction need not be tightly regulated. The part-units adjust accordingly to accommodate
whatever residual is left by the categorical-units.
The units form a rough hierarchy, with part-units on the bottom and categorical-units on the top.
Categorical-units receive strong recurrent connections, as shown in ﬁgure 3(c,d) implying that their
activity is more determined by other hidden units and less by the input (since the magnitude of the
input connections is bounded), and thus they are higher in the hierarchy. As shown in ﬁgure 7(c),
part-units receive most of their input from other part-units; categorical-units receive a larger fraction
of their input from other categorical-units. Whereas part-units have well-structured encoders and are
generally activated directly by the input on the ﬁrst iteration, categorical-units are more likely to ﬁrst
achieve a non-zero activation on the second iteration, as shown in ﬁgure 7(d), suggesting that they
require stimulation from part-units. The immediate response of part-units in contrast to the gradual
reﬁnement of categorical-units is apparent in ﬁgure 8, which shows the optimal decoding matrix for
selected units, inferred from their observed activity at each iteration.
4 Performance
The comparison of MNIST classiﬁcation performance in table 1 demonstrates the power of the hi-
erarchical representation learned by DrSAEs. Rather than learn to minimize the sum of equations 2
and 3, Gregor & LeCun (2010) train the LISTA encoder to approximate the code generated by a
traditional sparse coder. While they do not report classiﬁcation performance using LISTA, Gregor
and LeCun do evaluate MNIST classiﬁcation error using the related learned coordinate descent al-
gorithm. Sprechmann, Bronstein, & Sapiro (2012a,b) extend this approach by training a LISTA
auto-encoder to reconstruct the input directly, using loss functions similar to equation 2. Although
they identify the possibility of using regularization dependent upon supervised information, Sprech-
mann and colleagues do not consider a parameterized classiﬁer operating on a common hidden
8
(a)
 (b)
(c)
 (d)
(e)
Figure 7: Statistics of connections indicate the presence of a rough hierarchy, with categorical-units
on the top integrating over part-units on the bottom. Average explaining-away connection weight
Sij, binned by alignment between decoders, for connections from part-units to categorical-units (a).
If no units fall in a given bin, the average is set to zero. Average ﬁnal value of a unitzt=T
i , given that
zt=T
i >0, versus the angle between the encoder row Ei and decoder column Di (b). Average angle
between encoder row Ej and decoder column Dj of afferents to unit i, weighted by the strength
of the connection to unit i, versus the angle between encoder row Ei and decoder column Di (c).
Probability that z1
i = 0 and z2
i > 0, versus the angle between the encoder row Ei and decoder
column Di (d). Average value of the decoder column Di versus the angle between the encoder row
Ei and the decoder column Di (e).
representation. Instead, they train a separate encoder for each class, and classify each input based
upon the encoder with the lowest sparse coding error. DrSAEs signiﬁcantly outperform these other
techniques based upon a LISTA encoder.
DrSAEs also perform well compared to other techniques using encoders related to LISTA. Deep
sparse rectiﬁer neural networks (Glorot, Bordes, & Bengio, 2011) combine discriminative training
with an encoder similar to LISTA, but do not tie the parameters between the layers and only allow
the input to project to the ﬁrst layer. Differentiable sparse coding (Bradley & Bagnell, 2008) and
9
Enc Optimal inferred decoders Dec
(a)
(b)
Figure 8: Part-units (a) respond to the input quickly, while the activity of categorical-units (b) reﬁnes
slowly. Columns of the optimal decoding matrices Dt minimizing the input reconstruction error
||x −Dt ·zt||2
2 from the hidden representation zt for t= 1,...,T . The ﬁrst and last columns show
the corresponding encoder and decoder for the chosen representative units. Intermediate columns
represent successive iterations t.
LISTA auto-encoder,10 ×
(
289-1005)
3.76 (5.98 with 289 hidden units)
(Sprechmann, Bronstein, & Sapiro, 2012a)
Learned coordinate descent, 784-78450-10 2 .29
(Gregor & LeCun, 2010)
Differentiable sparse coding, 180-256∗-10 1 .30
(Bradley & Bagnell, 2008)
Deep sparse rectiﬁer neural network 1.20 (1.16 with tanh nonlinearity)
784-1000-1000-1000-10
(Glorot, Bordes, & Bengio, 2011)
Deep belief network, 784-500-500-2000-10 1 .18 (0.92 with dropout)
(Hinton, et al., 2012)
Discriminative recurrent sparse auto-encoder 1.08 (1.21 with 200 hidden units)
784-40011-10
Supervised dictionary learning, 1.05 (3.56 without contrastive loss)
45 ×(784-24∗) to 45 ×(784-96∗)
(Mairal, et al., 2009)
Table 1: MNIST classiﬁcation error rate (%) for pixel-permutation-agnostic encoders without
boosting-like augmentations. The ﬁrst column indicates the size of each layer in the speciﬁed en-
coder, separated by hyphens. Exponents specify the number of recurrent iterations; asterisks denote
repetition to convergence. 10×(···) indicates that a separate encoder is trained for each input class;
45 ×(···) indicates that a separate encoder is trained for each pairwise binary classiﬁcation prob-
lem. Further performance improvements have been reported with regularization techniques such as
dropout, architectures that enforce translation-invariance, and datasets augmented by deformations,
as discussed in the main text.
supervised dictionary learning (Mairal, et al., 2009) also train discriminatively, but effectively use
an inﬁnite-depth ISTA-like encoder, and are thus much less computationally efﬁcient than DrSAEs.
Supervised dictionary learning achieves performance statistically indistinguishable from DrSAEs
using a contrastive loss function. A similar technique achieves MNIST classiﬁcation error as low
10
as 0.54% when the dataset is augmented with shifted copies of the inputs (Mairal, Bach, & Ponce,
2012).
Additional regularizations and boosting-like techniques can further improve performance of net-
works with LISTA-like encoders. Recent examples include dropout, which trains and then averages
over a large set of random subnetworks formed by removing a constant fraction of the hidden units
from the original network (Goodfellow, et al., 2013; Hinton, et al., 2012). Deep belief networks
and deep Boltzmann machines ﬁne-tuned with dropout are the current state-of-the-art for pixel-
permutation-agnostic handwritten digit recognition (Hinton, et al., 2012), and can achieve MNIST
classiﬁcation error as low as 0.79% with a carefully tuned network structure and multi-step training
procedure. Deep convex networks, which iteratively reﬁne the classiﬁcation by successively training
a stack of classiﬁers, with the output of the i−1st classiﬁer provided as input to the ith classiﬁer,
can achieve an MNIST error of0.83% (Deng & Yu, 2011). Regularizing by explicit modeling of the
data manifold, and then minimizing the square of the Jacobian of the output along the tangent bun-
dle around the training datapoints, can reduce MNIST error to 0.81% (Rifai, et al., 2011). Further
performance improvements are possible if translation invariance is built directly into the network via
a convolutional architecture, and deformations of the inputs are included in the training set (LeCun,
et al., 1998), yielding error as low as0.23% (Ciresan, Meier, & Schmidhuber, 2012). These regular-
izations and augmentations are potentially compatible with DrSAE, but we defer their exploration
to future work.
Recurrence is essential to the performance of DrSAEs. If the number of recurrent iterations is de-
creased from eleven to two, MNIST classiﬁcation error in a network with 400 hidden units increases
from 1.08% to 1.32%. With only 200 hidden units, MNIST classiﬁcation error increases from1.21%
to 1.49%, although the hidden units still differentiate into part-units and categorical-units, as shown
in ﬁgure 3(d,f).
5 Discussion
It is widely believed that natural stimuli, such as images and sounds, fall near a low-dimensional
manifold within a higher-dimensional space (the manifold hypothesis) (Bengio, Courville, & Vin-
cent, 2012; Lee, Pedersen, & Mumford, 2003; Olshausen & Field, 2004). The low-dimensional data
manifold provides an intuitively compelling and empirically effective basis for classiﬁcation (Rifai,
et al., 2011; Simard, LeCun, & Denker, 1993; Simard, et al., 1998). The continuous deformations
that deﬁne the data manifold usually preserve identity, whereas even relatively small invalid trans-
formations may change the class of a stimulus. For instance, the various handwritten renditions of
the digit 3 in in the last column of ﬁgure 9(c) barely overlap, and so the Euclidean distance between
them in pixel space is greater than that to the nearest 8 formed by closing both loops. Neverthe-
less, smooth deformations of one 3 into another correspond to relatively short trajectories along
the data manifold,9 whereas the transformation of a 3 into an 8 requires a much longer path within
the data manifold. A prohibitive amount of data is required to fully characterize the data manifold
(Narayanan & Mitter, 2010), so it is often approximated by the set of linear submanifolds tangent to
the data manifold at the observed datapoints, known as the tangent spaces (Ekanadham, Tranchina,
& Simoncelli, 2011; Rifai, et al., 2011; Simard, et al., 1998). DrSAEs naturally and efﬁciently
form a tangent space-like representation, consisting of a point on the data manifold indicated by the
categorical-units, and a shift within the tangent space speciﬁed by the part-units.
Before discriminative ﬁne-tuning, DrSAEs perform a traditional part-based decomposition, familiar
from sparse coding, as shown in ﬁgure 9(a). The decoding matrix columns are class-independent,
local pen strokes, and many units make a comparable, small contribution to the reconstruction. Af-
ter discriminative ﬁne-tuning, the hidden units differentiate into sparse coding local part-units, and
global prototype categorical-units that integrate over them. As shown in ﬁgure 9(b,c), the input is de-
composed into a prototype, corresponding to a point on the data manifold; and a set of deformations
from this prototype along the data manifold, corresponding to shifts within the tangent space. The
same prototype can be used for very different inputs, as demonstrated in ﬁgure 9(c), since the space
of deformations is rich enough to encompass diverse transformations without moving off the data
9In particular, ﬁgure 9(c) shows how each input can be produced by identity-preserving deformations from
a common prototype, using the tangent space decomposition produced by our network.
11
Progressive reconstruction Fin Inp
(a)
(b)
(c)
Figure 9: Discriminative recurrent sparse auto-encoders decompose the input into a prototype and
deformations along the data manifold. The progressive reconstruction of selected inputs by the hid-
den representation before (a) or after (b,c) discriminative ﬁne-tuning. The columns from left to right
depict either the components of the reconstruction (top row of each pair), or the partial reconstruc-
tion induced by the ﬁrst n parts (bottom row of each pair). Parts are added to the reconstruction
in order of decreasing contribution magnitude; smoother transformations are possible with an op-
timized sequence. The last two columns show the ﬁnal reconstruction with all parts (Fin), and the
original input (Inp). Bars above the decoding matrix columns indicate the scale factor/hidden unit
activity associated with the column.
manifold. Even when the prototype is very different from the input, all steps along the reconstruction
trajectories in ﬁgure 9(b,c) are recognizable as members of the same class.
The prototypes learned by the categorical-units for each class are not simply the average over the
elements of the class, as depicted in ﬁgure 10. Each class includes many possible input variations,
so its average is blurry. The prototypes, in contrast, are sharp, and look like representative elements
of the appropriate class. Many categorical-units are available for each class, as shown in ﬁgure 6.
Not all categorical-units correspond to full prototypes; some capture global transformations of a
prototype, such as rotations (Simard, et al., 1998).
Consistent with prototypes for the non-negative MNIST inputs, the decoding matrix columns of
the categorical-units are generally positive, as shown in ﬁgure 7(e). In contrast, the decoders of
the part-units are approximately mean-zero and so cannot serve as prototypes themselves. Rather,
they shift and transform prototypes, moving activation from one region in the image to another, as
demonstrated in ﬁgure 9(b,c).
12
Avg Most class-speciﬁc units Avg Most class-speciﬁc units
Figure 10: The prototypes learned by categorical-units resemble representative instances of the ap-
propriate class, and are sharper than the average over all members of the class in the dataset. The
left-most column in each group depicts the average over all elements of each of the ten MNIST digit
classes. The other columns show the decoders of the associated units with the largest-magnitude
columns in the classiﬁcation matrix C. Bars above the decoders indicate the angle between the en-
coder and the decoder for the displayed unit. The most prototypical unit always makes the strongest
contribution to the classiﬁcation, and has a large (but not necessarily the largest) angle between its
encoder and decoder. Some units that make large contributions to the classiﬁcation represent global
transformations, such as rotations, of a prototype (Simard, et al., 1998).
Discrepancies between the prototype and the input due to transformations along the data manifold
are explained by class-consistent part-units, and only serve to further activate the categorical-units of
that class, as in ﬁgure 6(a,c). Discrepancies between the prototype and the input due to deformations
orthogonal to the data manifold are explained by class-incompatible part-units, and serve to suppress
the categorical-units of that class, both directly and via activation of incompatible categorical-units.
If the wrong prototype is turned on, the residual input will generally contain substantial unexplained
components. Part-units obey ISTA-like dynamics and thus function as a sparse coder on the residual
input, so part-units that match the unexplained components of the input will be activated. These part-
units will have positive connections to categorical-units with compatible prototypes, and so will tend
to activate categorical-units associated with the true class (so long as the unexplained components
of the input are diagnostic). The spuriously activated categorical-unit will not be able to sustain its
activity, since few compatible part-units will be required to capture the residual input.
The classiﬁcation approach used by DrSAEs is different from one based upon a traditional sparse
coding decomposition: it projects into the space of deviations from a prototype, which is not the
same as the space of prototype-free parts, as is clear from ﬁgure 9(a,b). For instance, a 5 can easily
be constructed using the parts of a6, making it difﬁcult to distinguish the two. Indeed, the ﬁrst seven
progressive reconstruction steps of the 6 in ﬁgure 9(a) could just as easily be used to produce a 5.
However, starting from a 6 prototype, the parts required to break the bottom loop are outside the
data manifold of the 6 class, and so will tend to change the active prototype.
DrSAEs naturally learn a hierarchical representation within a recurrent network, thereby implement-
ing a deep network with parameter sharing between the layers.
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Peer review:
[Review]:
Minor side comment: IN GENERAL, having a cost term at each iteration (time step of the unfolded network) does not eliminate the vanishing gradient problem!!!

The short-term dependencies can now be learned through the gradient on the cost on the early iterations, but the long-term effects may still be improperly learned. Now it may be that one is lucky (and that could apply in your setting) and that the weights that are appropriate for going from the state at t to a small cost at t+delta with small delta are also appropriate for minimizing the longer term costs for large delta. 

There are good examples of that in the literature. A toy example is the recurrent network that learns the parity of a sequence. Because of the recursive nature of the solution, if you do a very good job at predicting the parity for short sequences, there is a good chance that the solution will generalize properly to much longer sequences. Hence a curriculum that starts with short sequences and gradually extends to longer ones is able to solve the problem, where only training from long ones without intermediate targets at every time step completely fails.