Unsupervised Feature Learning for low-level Local Image Descriptors
Authors: Christian Osendorfer, Justin Bayer, Patrick van der Smagt
ICLR 2013

Abstract: Unsupervised feature learning has shown impressive results for a wide range of input modalities, in particular for object classification tasks in computer vision. Using a large amount of unlabeled data, unsupervised feature learning methods are utilized to construct high-level representations that are discriminative enough for subsequently trained supervised classification algorithms. However, it has never been quantitatively investigated yet how well unsupervised learning methods can find low-level representations for image patches without any supervised refinement. In this paper we examine the performance of pure unsupervised methods on a low-level correspondence task, a problem that is central to many Computer Vision applications. We find that a special type of Restricted Boltzmann Machines performs comparably to hand-crafted descriptors. Additionally, a simple binarization scheme produces compact representations that perform better than several state-of-the-art descriptors.

Unsupervised Feature Learning for low-level Local
Image Descriptors
Christian Osendorfer, Justin Bayer, Sebastian Urban, Patrick van der Smagt
Technische Universit¨at M¨unchen
{osendorf, bayerj, surban, smagt}@in.tum.de
Abstract
Unsupervised feature learning has shown impressive results for a wide range of in-
put modalities, in particular for object classiﬁcation tasks in computer vision. Us-
ing a large amount of unlabeled data, unsupervised feature learning methods are
utilized to construct high-level representations that are discriminative enough for
subsequently trained supervised classiﬁcation algorithms. However, it has never
been quantitatively investigated yet how well unsupervised learning methods can
ﬁnd low-level representationsfor image patches without any additional supervi-
sion. In this paper we examine the performance of pure unsupervised methods on
a low-level correspondence task, a problem that is central to many Computer Vi-
sion applications. We ﬁnd that a special type of Restricted Boltzmann Machines
(RBMs) performs comparably to hand-crafted descriptors. Additionally, a simple
binarization scheme produces compact representations that perform better than
several state-of-the-art descriptors.
1 Introduction
In this paper we tackle a recent computer vision dataset [2] from the viewpoint of unsupervised
feature learning. Why yet another dataset? There are already enough datasets that serve well for
evaluating feature learning algorithms. In particular for feature learning from image data, several
well-established benchmarks exist: Caltech-101 [10], CIFAR-10 [19], NORB [23], to name only a
few. Notably, these benchmarks are all object classiﬁcation tasks. Unsupervised learning algorithms
are evaluated by considering how well a subsequent supervised classiﬁcation algorithm performs
on high-level features that are found by aggregating the learned low-level representations [8]. We
think that mingling these steps makes it difﬁcult to assess the quality of the unsupervised algorithms.
A more direct way is needed to evaluate these methods, preferably where a subsequent supervised
learning step is completely optional.
We are not only at odds with the methodology of evaluating unsupervised learning algorithms. Gen-
eral object classiﬁcation tasks are always based on orientation- and scale-rectiﬁed pictures with
objects or themes ﬁrmly centered in the middle. We are looking for a dataset where it is possible
to show that unsupervised feature learning is beneﬁcial to the wide range of Computer Vision tasks
beyond object classiﬁcation, like tracking, stereo vision, panoramic stitching or structure from mo-
tion. One might argue, that object classiﬁcation acts as a good proxy for all these other tasks but
this hypothesis has not shown to be correct either theoretically or through empirical evidence. In-
stead, we chose the most general and direct task that can be be applied to low-level representations:
matching these representations, i.e. determining if two data samples are similar given their learned
representation.
Matching image descriptors is a central problem in Computer Vision, so hand-crafted descriptors
are always evaluated with respect to this task [28]. Given a dataset of labeled correspondences,
supervised learning approaches will ﬁnd representations and the accompanying distance metric that
1
arXiv:1301.2840v4  [cs.CV]  25 Apr 2013
are optimized with respect to the induced similarity measure. It is remarkable that hand-engineered
descriptors perform well under this task without the need to learn such a measurefor their represen-
tations in a supervised manner.
To the best of our knowledge it has never been investigated whether any of the many unsupervised
learning algorithms developed over the last couple of years can match this performance without
relying on any supervision signals. While we propose an additional benchmark for unsupervised
learning algorithms, we do not introduce a new learning algorithm. We rather investigate the per-
formance of the Gaussian RBM (GRBM) [39], its sparse variant (spGRBM) [29] and the mean
covariance RBM (mcRBM) [33] without any supervised learning with respect to the matching task.
As it turns out, the mcRBM performs comparably to hand-engineered feature descriptors. In fact
using a simple heuristic, the mcRBM produces acompact binarydescriptor that performs better than
several state-of-the-art hand-crafted descriptors.
We begin with a brief description of the dataset used for evaluating the matching task, followed by
a section on details of the training procedure. In section 4 we present our results, both quantitatively
and qualitatively and also mention other models that were tested but not further analyzed because
of overall bad performance. Section 5 concludes with a brief summary and an outlook for future
work. A review of GRBMs, spGRBMs and mcRBMs is provided in the appendix, section 6, for
completeness.
Related work Most similar in spirit to our work are [6, 20, 22]: Like us, [6, 22] are interested in
the behavior of unsupervised learning approaches without any supervised steps afterwards. Whereas
both investigate high-level representations. [20] learns a compact, binary representation with a
very deep autoencoder in order to do fast content-based image search ( semantic hashing, [36]).
Again, these representations are studied with respect to their capabilities to model high-level object
concepts. Additionally, various algorithms to learn high-level correspondences have been studied
[4, 37, 16] in recent years.
Finding (compact) low-level image descriptors should be an excellent machine learning task: Even
hand-designed descriptors have many free parameters that cannot (or should not) be optimized man-
ually. Given ground truth data for correspondences, the performance of supervised learning algo-
rithms is impressive [2]. Very recently, boosted learning with image gradient-based weak learners
has shown excellent results [43, 42] on the same dataset used in this paper. See section 2 of [43] for
more related work in the space of supervised metric learning.
2 Dataset
At the heart of this paper is a recently introduced dataset for discriminative learning of local image
descriptors [2]. It attempts to foster learning optimal low-level image representations using a large
and realistic training set of patch correspondences. The dataset is based on more than 1.5 million
image patches ( 64 ×64 pixels) of three different scenes: the Statue of Liberty (about 450,000
patches), Notre Dame (about 450,000 patches) and Yosemite’s Half Dome (about 650,000 patches).
The patches are sampled around interest points detected by Difference of Gaussians [27] and are
normalized with respect to scale and orientation 1. As shown in Figure 1, the dataset has a wide
variation in lighting conditions, viewpoints, and scales.
The dataset contains also approximately 2.5 million image correspondences. Correspondences be-
tween image patches are established via dense surface models obtained from stereo matching (stereo
matching, with its epipolar and multi-view constraints, is a much easier problem than unconstrained
2D feature matching). The exact procedure to establish correspondences is more involved and de-
scribed in detail in [2, Section II]. Because actual 3D correspondences are used, the identiﬁed 2D
patch correspondences show substantial perspective distortions resulting in a much more realistic
dataset than previous approaches [24, 28]. The dataset appears very similar to an earlier benchmark
of the same authors [47], yet the correspondences in the novel dataset resemble a much harder prob-
lem. The error rate at 95% detection of correct matches for the SIFT descriptor [27] raises from 6%
to 26%, the error rate for evaluating patch similarity in pixel space (using normalized sum squared
differences) raises from 20% to at least 48% (all numbers are take from [47] and [2] respectively),
1A similar dataset of patches centered on multi-scale Harris corners is also available.
2
Figure 1: Patch correspondences from the Liberty dataset. Note the wide variation in lighting,
viewpoint and level of detail. The patches are centered on interest points but otherwise can be
considered random, e.g. there is no reasonable notion of an object boundary possible. Figure taken
from [2].
for example. In order to facilitate comparison of various descriptor algorithms a large set of prede-
termined match/non-match patch pairs is provided. For every scene, sets comprising between 500
and 500,000 pairs (with 50% matching and 50% non-matching pairs) are available.
We don’t argue that this dataset subsumes or substitutes any of the previously mentioned bench-
marks. Instead, we think that it can serve to complement those. It constitutes an excellent testbed
for unsupervised learning algorithms: Experiments considering self-taught learning [32], effects of
semi-supervised learning, supervised transfer learning over input distributions with a varying de-
gree of similarity (the scenes of Statue of Liberty and Notredame show architectural structures,
while Half Dome resembles a typical natural scenery) and the effect of enhancing the dataset with
arbitrary image patches around keypoints can all be conducted in a controlled environment. Further-
more, end-to-end trained systems for (large) classiﬁcation problems (like [21, 5]) can be evaluated
with respect to this type of data distribution and task.
3 Training Setup
Different to [2], our models are trained in an unsupervised fashion on the available patches. We train
on one scene (400,000 randomly selected patches from this scene) and evaluate the performance on
the test set of every scene. This allows us to investigate the self-taught learning paradigm [32]. We
also train on all three scenes jointly (represented by 1.2 million image patches) and then evaluate
again every scene individually.
3.1 GRBM/spGRBM
The GRBM and spGRBM (see Appendix, section 6.2) only differ in the setting of the sparsity
penalty λsp, all other settings are the same. We use CD1 [13] to compute the approximate gradient
of the log-likelihood and the recently proposed rmsprop [41] method as gradient ascent method.
Compared to standard minibatch gradient ascent, we ﬁnd that rmsprop is a more efﬁcient method
with respect to the training time necessary to learn good representations: it takes at most half of the
training time necessary for standard minibatch gradient ascent.
Before learning the parameters, we ﬁrst scale all image patches to16×16 pixels. Then we preprocess
all training samples by subtracting the vectors’ mean and dividing by the standard deviation of its
elements. This is a common practice for visual data and corresponds to local brightness and contrast
normalization. [39, Section 2.2] gives also a theoretical justiﬁcation for why this preprocessing step
is necessary to learn a reasonable precision matrix Λ. We ﬁnd that this is the only preprocessing
scheme that allows GRBM and spGRBM to achieve good results. In addition, it is important to learn
3
Λ—setting it to the identity matrix, a common practice [14], also produces dissatisfying error rates.
Note that originally it was considered that learning Λ is mostly important when one wants to ﬁnd a
good density (i.e. generative) model of the data.
Both GRBM and spGRBM have 512 hidden units. The elements of W are initialized according to
N(0,0.1), the biases are initialized to 0. rmsprop uses a learning rate of 0.001, the decay factor is
0.9, the minibatch size is 128. We train both models for 10 epochs (this takes about 15 minutes on
a consumer GPU for 400000 patches). For the spGRBM we use a sparsity target of ρ= 0.05 and a
sparsity penalty of λsp = 0.2. spGRBM is very sensitive to settings of λsp [38]—setting it too high
results in dead representations (samples that have no active hidden units) and the results deteriorate
drastically.
3.2 mcRBM
mcRBM (see Appendix, section 6.3) training is performed using the code from [33]. We resam-
ple the patches to 16 ×16 pixels. Then the samples are preprocessed by subtracting their mean
(patchwise), followed by PCA whitening, which retains 99% of the variance. The overall training
procedure (with stochastic gradient descent) is identical to the one described in [33, Section 4]. We
train all architectures for a total of 100 epochs, however updating P is only started after epoch
50. We consider two different mcRBM architectures: The ﬁrst has 256 mean units, 512 factors
and 512 covariance units. P is not constrained by any ﬁxed topography. We denote this architec-
ture by mcRBM(256,512/512). The second architecture is concerned with learning more compact
representations: It has 64 mean units, 576 factors and 64 covariance units. P is initialized with a
two-dimensional topography that takes 5 ×5 neighborhoods of factors with a stride equal to 3. We
denote this model by mcRBM(64,576/64). On a consumer grade GPU it takes 6 hours to train the
ﬁrst architecture on 400000 samples and 4 hours to train the second architecture on the same number
of samples.
4 Results
For the results presented in this section (Table 1) we follow the evaluation procedure of [2]: For
every scene (Liberty (denoted by LY), Notredame (ND) and Half Dome (HD)), we use the labeled
dataset with 100,000 image pairs to assess the quality of a trained model on this scene. In order to
save space we do not present ROC curves and only show the results in terms of the 95% error rate
which is the percent of incorrect matches when 95% of the true matches are found: After computing
the respective distances for all pairs in a test set, a threshold is determined such that 95% of all
matching pairs have a distance below this threshold. Non-matching pairs with a distance below this
threshold are considered incorrect matches.
Table 1 consists of two subtables. Table 1a presents the error rates for GRBM, spGRBM and
mcRBM when no limitations on the size of representations are placed. Table 1b only considers
descriptors that have an overall small memory footprint. For GRBM and spGRBM we use the
activations of the hidden units given a preprocessed input patch v as descriptor D(v) (see eq. 5,
section 6.1):
D(v) =σ(vT Λ
1
2 W + b)
For the mcRBM a descriptor is formed by using the activations of thelatent covariance units alone,
see eq. 8, section 6.3:
D(v) =σ(PT (CT v)2 + c)
This is in accordance with manually designed descriptors. Many of these rely on distributions (i.e.
histograms) of intensity gradients or edge directions [27, 28, 1], structural information which is
encoded by the covariance units (see also [35, Section 2])2.
4.1 Distance metrics
As we explicitly refrain from learning a suitable (with respect to the correspondence task) distance
metric with a supervised approach, we have to resort to standard distance measures. The Euclidean
2Extending the descriptor with mean units degrades results.
4
Test set
Method Training set LY ND HD
SIFT – 28.1 20.9 24.7
LY 47.6 33.5 41.4
GRBM ND 50.0 33.4 42.5
(L1ℓ1) HD 49.0 34.0 41.5
LY/ND/HD 48.7 33.5 42.1
LY 37.9 26.9 34.3
spGRBM ND 40.0 28.0 35.4
(L1ℓ1) HD 39.1 27.9 34.9
LY/ND/HD 37.5 26.6 33.6
LY 31.3 25.1 34.5
mcRBM ND 34.0 25.6 33.0
(L1ℓ2) HD 31.2 22.3 25.7
LY/ND/HD 30.8 24.8 33.3
LY 34.7 24.2 38.6
mcRBM ND 33.3 24.8 44.9
(JSD) HD 29.9 22.7 37.6
LY/ND/HD 30.0 23.1 39.8
(a)
Test set
Method Training set LY ND HD
SIFT – 31.7 22.8 25.6
BRIEF – 59.1 54.5 54.9
BRISK – 79.3 74.8 73.2
SURF – 54.0 45.5 43.5
LY – 16.9 22.8BinBoost ND 20.4 – 18.9(8 bytes) HD 21.6 14.5 –
LY – 31.1 34.4ITQ-SIFT ND 37.0 – 34.3(8 bytes) HD 37.3 30.5 –
LY – 43.1 47.2D-Brief ND 46.2 – 51.3(4 bytes) HD 53.3 43.9 –
LY 36.2 39.9 64.9
mcRBM ND 46.2 34.5 56.1
(8 bytes) YM 43.4 37.4 53.0
LY/ND/HD 40.5 36.6 55.4
(b)
Table 1: Error rates, i.e. the percent of incorrect matches when 95% of the true matches are found.
All numbers for GRBM, spGRBM and mcRBMs are given within±0.5%. Every subtable, indicated
by an entry in the Method column, denotes a descriptor algorithm. Descriptor algorithms that do not
require learning (denoted by – in the columnTraining set) are represented by one line. The numbers
in the columns labeled LY , ND and HD are the error rates of a method on the respective test set for
this scene. Supervised algorithms are not evaluated (denoted by –) on the scene they are trained on.
The Training setLY/ND/HD encompasses 1.2 million patches of all three scenes; this setting is only
possible for unsupervised learning methods. (a) Error rates for several unsupervised algorithms
without restricting the size of the learned representation. GRBM, spGRBM and mcRBM learn
descriptors of dimensionality 512. (L1ℓ1) denotes that the error rates for a method are with respect
to ℓ1 normalization of the descriptor under the L1 distance. (b) Results for compact descriptors.
BRIEF (32 bytes) [3] and BRISK (64 bytes) [25] are binary descriptors, SURF [1] is a real valued
descriptor with 64 dimensions. BinBoost [42], ITQ-SIFT [12] and D-Brief [44] learn compact
binary descriptors with supervision. Numbers for BRIEF, BRISK, SURF, BinBoost and ITQ-SIFT
are from [42].
distance is widely used when comparing image descriptors. Yet, considering the generative nature of
our models we follow the general argumentation of [17] and choose the Manhattan distance, denoted
in this text by L1. We also consider two normalization schemes for patch representations, ℓ1 and ℓ2
(i.e. after a feature vector x is computed, its length is normalized such that ∥x∥1 = 1or ∥x∥2 = 1).
Given a visible input both (sp)GRBM and mcRBM compute features that resemble parameters of
(conditionally) independent Bernoulli random variables. Therefore we consider the Jensen-Shannon
divergence (JSD) [26] as an alternative similarity measure. Finally, for binary descriptors, we use
the Hamming distance.
4.2 SIFT Baseline
SIFT [27] (both as interest point detector and descriptor) was a landmark for image feature matching.
Because of its good performance it is one of the most important basic ingredients for many different
kinds of Computer Vision algorithms. It serves as a baseline for evaluating our models. We use
vlfeat [45] to compute the SIFT descriptors.
5
The performance of the SIFT descriptor, ℓ1-normalized, is reported (using L1 distance) in Table 1a,
ﬁrst entry. ℓ1 normalization provides better results than ℓ2 normalization or no normalization at all.
SIFT performs descriptor sampling at a certain scale relative to the Difference of Gaussians peak.
In order to achieve good results, it is essential to optimize this scale parameter [2, Figure 6] on
every dataset. Table 1b is concerned with evaluating compact descriptors: the ﬁrst entry shows the
performance of SIFT when used as a 128-byte descriptor (i.e. no normalization applied, but again
optimized for the best scale parameter) with L1 distance.
4.3 Quantitative analysis
Table 1a shows that SIFT performs better than all three unsupervised methods. mcRBM(256,
512/512) performs similar to SIFT when trained on Half Dome, albeit at the cost of a 4.5 times
larger descriptor representation. The compact binary descriptor (the simple binarization scheme
is described below) based on mcRBM(64, 576/64) performs remarkably well, comparable or even
better than several state-of-the-art descriptors (either manually designed or trained in a supervised
manner), see Table 1b, last entry. We discuss in more detail several aspects of the results in the
following paragraphs.
GRBM and spGRBM spGRBM performs considerably better than its non-sparse version (see
Table 1a, second and third entries). This is not necessarily expected: Unlike e.g. in classiﬁcation
[8] sparse representations are considered problematic with respect to evaluating distances directly.
Lifetime sparsitymay be after all beneﬁcial in this setting compared to strictly enforced population
sparsity. We plan to investigate this issue in more detail in future work by comparing spGRBM to
Cardinality restricted boltzman machines [38] on this dataset.
Self-taught paradigm We would expect that the performance of a model trained on the Liberty
dataset and evaluated on the Notre Dame scene (and vice versa) should be noticeably better than
the performance of a model trained on Half Dome and evaluated on the two architectural datasets.
However, this is not what we observe. In particular for the mcRBM (both architectures) it is the
opposite: Training on the natural scene data leads to much better performance than the assumed
optimal setting.
Jensen-Shannon Divergence Both GRBM and spGRBM perform poorly under the Jensen-
Shannon divergence similarity (overall error rates are around 60%), therefore we don’t report these
numbers in the table. Similar, results for mcRBM under JSD are equally bad. However, if onescales
down P by a constant (we found the value of 3 appropriate), the results with respect to JSD improve
noticeably, see Table 1a, the last entry. The performance on the Half Dome dataset is still not good
– the scaling factor should be learned [9], which we also plan for future work.
Compact binary descriptor We were not successful in ﬁnding a good compact representa-
tion with either GRBM or spGRBM. Finding compact representations for any kind of input
data should be done with multiple layers of nonlinearities [20]. But even with only two layers
(mcRBM(64,576/64)) we learn relatively good compact descriptors. If features are binarized, the
representation can be made even more compact (64 bits, i.e. 8 bytes). In order to ﬁnd a suitable
binarization threshold we employ the following simple heuristic: After training on a dataset is ﬁn-
ished we histogram all activations(values between 0 and 1) of the training setand use the median
of this histogram as the threshold.
4.4 Qualitative analysis
We brieﬂy comment on the developed ﬁlters (Figure 2). Unsurprisingly, spGRBM (Figure 2a) and
mcRBM (Figure 2b—these are columns from C) learn Gabor like ﬁlters. At a closer look we make
some interesting observations: Figure 2c shows the diagonal elements of Λ
1/2
from a spGRBM.
When computing a latent representation, the input v is scaled (elementwise) by this matrix, which,
visualized as a 2D image, resembles a Gaussian that is dented at the center, the location of the
keypoint of every image patch. The mcRBM also builds ﬁlters around the keypoint: Figure 2d shows
some unusual ﬁlters from C. They are centered around the keypoint and bear a strong resemblance
to discriminative projections (Figure 2e) that are learned in a supervised way on this dataset [2,
6
(a)
 (b)
(c)
 (d)
 (e)
(f)
Figure 2: (a) Typical ﬁlters learned with spGRBM. (b) Filters from an mcRBM. (c) The pixelwise
inverted standard deviations learned with a spGRBM plotted as a 2D image (darker gray intensities
resemble lower numerical values). An input patch is elementwise multiplied with this image when
computing the latent representation. This ﬁgure is generated by training on 32 ×32 patches for
better visibility, but the same qualitative results appear with 16 ×16 patches. (d) The mcRBM also
learns some variants of log-polar ﬁlters centered around the DoG keypoint. These are very similar
to ﬁlters found when optimizing for the correspondence problem in a supervised setting. Several
of such ﬁlters are shown in subﬁgure (e), taken from [2, Figure 5]. Finally (f), the basic keypoint
ﬁlters are combined with Garbor ﬁlters, if these are placed close to the center; the Garbor ﬁlters get
systematically arranged around the keypoint ﬁlters.
Figure 5]. Qualitatively, the ﬁlters in Figure 2d resemble log-polar ﬁlters that are used in several
state-of-the-art feature designs [28]. The very focused keypoint ﬁlters (ﬁrst column in Figure 2d)
are often combined with Gabor ﬁlters placed in the vicinity of the center– the Garbor ﬁlters appear
on their own, if they are too far from the center. If an mcRBM is trained with a ﬁxed topography for
P, one sees that the Gabor ﬁlters get systematically arranged around the keypoint (Figure 2f).
4.5 Other models
We also trained several other unsupervised feature learning models: GRBM with nonlinear rectiﬁed
hidden units3 [30], various kinds of autoencoders (sparse [7] and denoising [46] autoencoders), K-
3Our experiments indicate that rmsprop is in this case also beneﬁcial with respect to the ﬁnal results: It
learns models that perform about 2-3% better than those trained with stochastic gradient descent.
7
means [7] and two layer models (stacked RBMs, autoencoders with two hidden layers, cRBM [34]).
None of these models performed as good as the spGRBM.
5 Conclusion
We start this paper suggesting that unsupervised feature learning should be evaluated (i) without us-
ing subsequent supervised algorithms and (ii) more directly with respect to its capacity to ﬁnd good
low-level image descriptors. A recently introduced dataset for discriminatively learning low-level
local image descriptors is then proposed as a suitable benchmark for such an evaluation scheme that
complements nicely the existing benchmarks. We demonstrate that an mcRBM learns real-valued
and binary descriptors that perform comparably or even better to several state-of-the-art methods on
this dataset.
In future work we plan to evaluate deeper architectures [20], combined with sparse convolutional
features [18] on this dataset. Moreover, ongoing work investigates several algorithms [4, 37] for
supervised correspondence learning on the presented dataset.
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6 Appendix
6.1 Gaussian-Binary Restricted Boltzmann Machine
The Gaussian-Binary Restricted Boltzmann Machine (GRBM) is an extension of the Binary-Binary
RBM [11] that can handle continuous data [15, 39]. It is a bipartite Markov Random Field over a
set of visible units, v ∈RNv , and a set of hidden units, h ∈{0,1}Nh . Every conﬁguration of units
v and units h is associated with an energy E(v,h), deﬁned as
E(v,h; θ) =1
2vT Λv −vT Λa −hT b −vT ΛWh (1)
with θ = (W ∈RNv×Nh ,a ∈RNv ,b ∈RNh ,Λ ∈RNv×Nv ), the model parameters. W rep-
resents the visible-to-hidden symmetric interaction terms, a and b represent the visible and hidden
biases respectively and Λ is the precision matrix of v, taken to be diagonal. E(v,h) induces a
probability density function over v and h:
p(v,h; θ) =exp
(
−E(v,h; θ)
)
Z(θ) (2)
where Z(θ) is the normalization partition function, Z(θ) =
∫∑
h exp
(
−E(v,h; θ)
)
dv.
Learning the parameters θ is accomplished by gradient ascent in the log-likelihood of θ given N
i.i.d. training samples. The log-probability of one training sample is
log p(v) =−1
2vT Λv + vT Λa +
Nh∑
j
log
(
1 + exp
(Nv∑
i
vT
i (Λ
1
2 W)ij + bj
))
−Z(θ) (3)
Evaluating Z(θ) is intractable, therefore algorithms like Contrastive Divergence (CD) [13] or per-
sistent CD (PCD) [40] are used to compute an approximation of the log-likelihood gradient. The
bipartite nature of an (G)RBM is an important aspect when using these algorithms: The visible units
are conditionally independent given the hidden units. They are distributed according to a diagonal
Gaussian:
p(v |h) ∼N(Λ−1
2 Wh + a,Λ−1) (4)
Similarly, the hidden units are conditionally independent given the visible units. The conditional
distribution can be written compactly as
p(h |v) =σ(vT Λ
1
2 W + b) (5)
where σdenotes the element-wise logistic sigmoid function, σ(z) = 1/(1 +e−z).
6.2 Sparse GRBM
In many tasks it is beneﬁcial to have features that are only rarely active [29, 8]. Sparse activation of a
binary hidden unit can be achieved by specifying a sparsity targetρand adding an additional penalty
term to the log-likelihood objective that encourages the actual probability of unit j of being active,
qj, to be close to ρ[29, 14]. This penalty is proportional to the negative KL divergence between the
hidden unit marginal qj = 1
N
∑
n p(hj = 1|vn) and the target sparsity:
λsp
(
ρlog qj + (1−ρ) log(1−qj)
)
, (6)
where λsp represents the strength of the penalty. This term enforces sparsity of feature j over the
training set, also referred to aslifetime sparsity. The hope is that the features for one training sample
are then encoded by a sparse vector, corresponding topopulation sparsity. We denote a GRBM with
a sparsity penalty λsp >0 as spGRBM.
10
6.3 Mean-Covariance Restricted Boltzmann Machine
In order to model pairwise dependencies of visible units gated by hidden units, a third-order RBM
can be deﬁned with a weight wijk for each triplet vi,vj,hk. By factorizing and tying these weights,
parameters can be reduced to a ﬁlter matrix C ∈RNv×F and a pooling matrix P ∈RF×Nh . C
connects the input to a set of factors and P maps factors to hidden variables. The energy function
for this cRBM [34] is
Ec(v,hc; θ) =−(vT CT )2Phc −cT hc (7)
where (·)2 denotes the element-wise square operation and θ = {C,P,c}. Note that P has to be
non-positive [34, Section 5]. The hidden units of the cRBM are still conditionally independent given
the visible units, so inference remains simple. Their conditional distribution (given visible state v)
is
p(hc |v) =σ(PT (CT v)2 + c) (8)
The visible units are coupled in a Markov Random Field determined by the setting of the hidden
units:
p(v |hc) ∼N(0,Σ) (9)
with
Σ−1 = Cdiag(−Phc)CT (10)
As equation 9 shows, the cRBM can only model Gaussian inputs with zero mean. For general
Gaussian-distributed inputs the cRBM and the GRBM can be combined into the mean-covariance
RBM (mcRBM) by simply adding their respective energy functions:
Emc(v,hm,hc; θ,θ′) =Em(v,hm; θ) +Ec(v,hc,θ′) (11)
Em(v,hm; θ) denotes the energy function of the GRBM (see eq. 1) with Λ ﬁxed to the identity
matrix. The resulting conditional distribution over the visible units, given the two sets of hidden
units hm (mean units) and hc (covariance units) is
p(v |hm,hc) ∼N(ΣWhm,Σ) (12)
with Σ deﬁned as in eq. 10. The conditional distributions p(hm|v) and p(hc|v) are still as in eq. 5
and eq. 7 respectively. The parameters θ,θ′ can again be learned using approximate Maximum
Likelihood Estimation, e.g. via CD or PCD. These methods require to sample from p(v|hm,hc),
which involves an expensive matrix inversion (see eq. 10). Instead, samples are obtained by using
Hybrid Monte Carlo (HMC) [31] on the mcRBM free energy [33].
11


Peer review:
[Review]:
his paper proposes a dataset to benchmark the correspodence problem
in computer vision. The dataset consists of image patches that have
groundtruth matching pairs (using separate algorithms). Extensive
experiments show that RBMs perform well compared to hand-crafted
features.

I like the idea of using itermediate evaluation metrics to measure the
progress of unsupervised feature learning and deep learning. That
said, comparing the methods on noisy groundtruth (results of other
algorithms) may have some bias.

The experiments could be made stronger if algorithms such as
Autoencoders or Kmeans (Coates et al, 2011, An Analysis of
Single-Layer Networks in Unsupervised Feature Learning) are
considered.

If we can consider the groundtruth as clean, will supervised learning
a deep (convolutional) network using the groundtruth produce better
results?