Proximal Methods for Sparse Hierarchical Dictionary Learning.

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Proximal Methods for Sparse Hierarchical Dictionary Learning
Rodolphe Jenatton1
Julien Mairal1
Guillaume Obozinski
Francis Bach
RODOLPHE.JENATTON@INRIA.FR
JULIEN.MAIRAL@INRIA.FR
GUILLAUME.OBOZINSKI@INRIA.FR
FRANCIS.BACH@INRIA.FR
INRIA - WILLOW Project, Laboratoire d’Informatique de l’Ecole Normale Sup´ erieure (INRIA/ENS/CNRS UMR 8548)
23, avenue d’Italie, 75214 Paris. France
Abstract
We propose to combine two approaches for mod-
eling data admitting sparse representations: on
the one hand, dictionary learning has proven ef-
fective for various signal processing tasks. On
the other hand, recent work on structured spar-
sity provides a natural framework for modeling
dependencies between dictionary elements. We
thus consider a tree-structured sparse regulariza-
tion to learn dictionaries embedded in a hierar-
chy.The involved proximal operator is com-
putable exactly via a primal-dual method, allow-
ing the use of accelerated gradient techniques.
Experiments show that for natural image patches,
learned dictionary elements organize themselves
in such a hierarchical structure, leading to an im-
proved performance for restoration tasks. When
applied to text documents, our method learns hi-
erarchies of topics, thus providing a competitive
alternative to probabilistic topic models.
1. Introduction
Learned sparse representations, initially introduced by
Olshausen & Field (1997), have been the focus of much
research in machine learning and signal processing, lead-
ing notably to state-of-the-art algorithms for several prob-
lems in image processing (Elad & Aharon, 2006). Model-
ing signals as a linear combination of a few “basis” vectors
offers more flexibility than decompositions based on prin-
cipal component analysis and its variants. Indeed, sparsity
allows for overcomplete dictionaries, whose number of ba-
sis vectors are greater than the original signal dimension.
1Contributed equally.
Appearing in Proceedings of the 27thInternational Conference
on Machine Learning, Haifa, Israel, 2010. Copyright 2010 by the
author(s)/owner(s).
As far as we know, while much attention has been given
to efficiently solving the corresponding optimization prob-
lem (Lee et al., 2007; Mairal et al., 2010), there are few at-
tempts in the literature to make the model richer by adding
structure between dictionary elements (Bengio et al., 2009;
Kavukcuoglu et al., 2009). We propose to use recent work
on structured sparsity (Zhao et al., 2009; Jenatton et al.,
2009; Kim & Xing, 2009) to embed the dictionary ele-
ments in a hierarchy.
Hierarchies of latent variables, typically used in neural net-
works and deep learning architectures (see Bengio, 2009
and references therein) have emerged as a natural structure
in several applications, notably to model text documents.
Indeed, inthecontextoftopicmodels(Blei et al.,2003), hi-
erarchical models using Bayesian non-parametric methods
have been proposed by Blei et al. (2010). Quite recently,
hierarchies have also been considered in the context of ker-
nel methods (Bach, 2009). Structured sparsity has been
used to regularize dictionary elements by Jenatton et al.
(2010), but to the best of our knowledge, it has never been
used to model dependencies between them.
This paper makes three contributions:
• We propose to use a structured sparse regularization to
learn a dictionary embedded in a tree.
• We show that the proximal operator for a tree-structured
sparse regularization can be computed exactly in a finite
number of operations using a primal-dual approach, with a
complexity linear, or close to linear, in the number of vari-
ables. Accelerated gradient methods (e.g., Nesterov, 2007)
can then be applied to solve tree-structured sparse decom-
position problems, which may be useful in other uses of
tree-structured norms (Kim & Xing, 2009; Bach, 2009).
• Our method establishes a bridge between dictionary
learning for sparse coding and hierarchical topic models
(Blei et al., 2010), which builds upon the interpretation of
topic models as multinomial PCA (Buntine, 2002), and can
learn similar hierarchies of topics. See Section 5 for a dis-
cussion.

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Proximal Methods for Sparse Hierarchical Dictionary Learning
2. Problem Statement
2.1. Dictionary Learning
Let us consider a set X = [x1,...,xn] ∈ Rm×nof n
signals of dimension m. Dictionary learning is a matrix
factorization problem that aims to represent these signals
as linear combinations of dictionary elements, denoted here
by the columns of a matrix D = [d1,...,dp] ∈ Rm×p.
More precisely, the dictionary D is learned along with a
matrix of decomposition coefficients A = [α1,...,αn] ∈
Rp×n, so that xi≈ Dαifor every signal xi, as measured
by any convex loss, e.g., the square loss in this paper.
While learning simultaneously D and A, one may want
to encode specific prior knowledge about the task at
hand, such as, for example, the positivity of the de-
composition (Lee & Seung, 1999), or the sparsity of A
(Olshausen & Field, 1997; Lee et al., 2007; Mairal et al.,
2010). This leads to penalizing or constraining (D,A) and
results in the following formulation:
min
D∈D,A∈A
1
n
n
?
i=1
?1
2||xi− Dαi||2
2+ λΩ(αi)
?
,
(1)
where A and D denote two convex sets and Ω is a regu-
larization term, usually a norm, whose effect is controlled
by the regularization parameter λ > 0. Note that D is as-
sumed to be bounded to avoid any degenerate solutions of
Eq. (1). For instance, the standard sparse coding formula-
tion takes Ω to be the ℓ1norm, D to be the set of matrices
in Rm×pwhose columns are in the unit ball of the ℓ2norm,
with A = Rp×n(Lee et al., 2007; Mairal et al., 2010).
However, this classical setting treats each dictionary ele-
ment independently from the others, and does not exploit
possible relationships between them. We address this po-
tential limitation of the ℓ1norm by embedding the dictio-
nary in a tree structure, through a hierarchical norm intro-
duced by Zhao et al. (2009) and Bach (2009), which we
now present.
2.2. Hierarchical Sparsity-Inducing Norms
We organize the dictionary elements in a rooted-tree
T composed of p nodes, one for each dictionary ele-
ment dj, j ∈ {1,...,p}. We will identify these indices
j in {1,...,p} and the nodes of T . We want to exploit the
structure of T in the following sense: the decomposition
of any vector x can involve a dictionary element djonly if
the ancestors of djin T are themselves part of the decom-
position. Equivalently, one can say that when a dictionary
element djis not involved in the decomposition of a vec-
tor x then its descendants in T should not be part of the
decomposition. While these two views are equivalent, the
latter leads to an intuitive penalization term.
Figure 1. Left: example of a tree-structured set of groups G
(dashed contours in red), corresponding to a tree T = {1,...,6}
(in black).Right, example of a sparsity pattern: the groups
{2,4},{4} and {6} are set to zero, so that the corresponding
nodes (in gray) that form subtrees of T are removed. The remain-
ing nonzero variables {1,3,5} are such that, if a node is selected,
the same goes for all its ancestors.
To obtain models with the desired property, one considers
for all j in T , the group gj ⊆ {1,...,p} of dictionary
elements that only contains j and all its descendants, and
penalizes the number of such groups that are involved in
the decomposition of x (a group being “involved in the de-
composition” meaning here that at least one of its dictio-
nary element is part of the decomposition). We call G this
set of groups (Figure 1).
While this penalization is non-convex, a convex proxy has
been introduced by Zhao et al. (2009) and was further con-
sidered by Bach (2009) and Kim & Xing (2009) in the con-
text of regression. For any vector α ∈ Rp, let us define
Ω(α)
△=
?
g∈G
wg?α|g?,
where α|g is the vector of size p whose coordinates are
equal to those of α for indices in the set g, and 0 other-
wise2. ?.? stands either for the ℓ∞or ℓ2norm, and (wg)g∈G
denotes some positive weights3. As analyzed by Zhao et al.
(2009), when penalizing by Ω, some of the vectors α|gare
set to zero for some g ∈ G. Therefore, the components
of α corresponding to some entire subtrees of T are set to
zero, which is exactly the desired effect (Figure 1).
Note that even though we have presented for simplicity rea-
sons this hierarchical norm in the context of a single tree
with a single element at each node, it can be extended eas-
ily to the case of forests of trees, and/or trees containing
several dictionary elements at each node. More generally,
this formulation can be extended with the notion of tree-
structured groups, which we now present.
2Note the difference with the notation αg, which is often used
in works on structured sparsity, where αgis a vector of size |g|.
3For a complete definition of Ω for any ℓq norm, a discussion
of the choice of q, and a strategy for choosing the weights wg, see
(Zhao et al., 2009; Kim & Xing, 2009).

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Proximal Methods for Sparse Hierarchical Dictionary Learning
Definition 1 (Tree-structured set of groups.)
groups G = {g}g∈G is said to be tree-structured in
{1,...,p}, if?
(g∩h ?= ∅) ⇒ (g ⊆ h or h ⊆ g). For such a set of groups,
there exists a (non-unique) total order relation ? such that:
A set of
g∈Gg = {1,...,p} and for all g,h ∈ G,
g ? h ⇒
?g ⊆ h or g ∩ h = ∅?.
Sparse hierarchical norms having been introduced, we now
address the optimization dealing with such norms.
3. Optimization
Optimization for dictionary learning has already been in-
tensively studied, and a typical scheme alternating between
the variables D and A = [α1,...,αn], i.e., minimizing
over one while keeping the other one fixed, yields good re-
sults in general (Lee et al., 2007). The main difficulty of
our problem lies essentially in the optimization of the vec-
tors αi, i ∈ {1,...,n} for D fixed, since n may be large,
and since it requires to deal with the nonsmooth regulariza-
tion term Ω. The optimization of the dictionary D (for A
fixed) is in general easier, as discussed in Section 3.5.
Within the context of regression, several optimization
methods to cope with Ω have already been proposed. A
boosting-like technique with a path-following strategy is
used by Zhao et al. (2009).
reweighted least-squares scheme when ?.? is the ℓ2norm.
The same approach is considered by Bach (2009), but built
upon an active set strategy. In this paper, we propose to
perform the updates of the vectors αibased on a proximal
approach which we now introduce.
Kim & Xing (2009) uses a
3.1. Proximal Operator for the Norm Ω
Proximal methods have drawn increasing attention in the
machine learning community (e.g., Ji & Ye, 2009 and ref-
erences therein), especially because of their convergence
rates (optimal for the class of first-order techniques) and
their ability to deal with large nonsmooth convex problems
(e.g., Nesterov, 2007; Beck & Teboulle, 2009). In a nut-
shell, these methods can be seen as a natural extension of
gradient-based techniques when the objective function to
minimize has a nonsmooth part. In our context, when the
dictionary D is fixed and A = Rp, we minimize for each
signal x the following convex nonsmooth objective func-
tion w.r.t. α ∈ Rp:
f(α) + λΩ(α),
where f(α) stands for the data-fitting term1
At each iteration of the proximal algorithm, f is linearized
around the current estimate ˆ α, and the current value of α
is updated as the solution of the proximal problem:
2?x − Dα?2
2.
min
α∈Rpf(ˆ α)+(α − ˆ α)⊤∇f(ˆ α) + λΩ(α) +L
2?α − ˆ α?2
2.
The quadratic term keeps the update in a neighborhood
where f is close to its linear approximation, and L > 0
is a parameter. This problem can be rewritten as,
min
α∈Rp
1
2?α −?ˆ α −1
L∇f(ˆ α)??2
2+λ
LΩ(α).
Solving efficiently and exactly this problem is crucial to en-
joy the fast convergence rates of proximal methods. In ad-
dition, when the nonsmooth term Ω is not present, the pre-
vious proximal problem exactly leads to the standard gra-
dient update rule. More generally, the proximal operator
associated with our regularization term λΩ, is the function
that maps a vector u ∈ Rpto the (unique) solution of
min
v∈Rp
1
2?u − v?2
2+ λΩ(v).
(2)
In the simpler setting where G is the set of singletons, Ω is
the ℓ1norm, and the proximal operator is the (elementwise)
soft-thresholding operator uj ?→ sign(uj)max(|uj| −
λ,0), j ∈ {1,...,p}. Similarly, when the groups in G
form a partition of the set of variables, we have a group
Lasso like penalty, and the proximal operator can be com-
puted in closed-form (see Bengio et al., 2009 and refer-
ences therein). This is a priori not possible anymore as
soon as some groups in G overlap, which is always the case
in our hierarchical setting with tree-structured groups.
3.2. Primal-Dual Interpretation
We now show that Eq. (2) can be solved with a primal-
dual approach. The procedure solves a dual formulation of
Eq. (2) involving the dual norm4of ?.?, denoted by ?.?∗,
and defined by ?κ?∗= max?z?≤1z⊤κ for any vector κ
in Rp. The formulation is described in the following lemma
that relies on conic duality (Boyd & Vandenberghe, 2004):
Lemma 1 (Dual of the proximal problem)
Let u ∈ Rpand let us consider the problem
max
ξ∈Rp×|G|−1
2
??u −
?
g∈G
ξg?2
2− ?u?2
2
?
s.t. ∀g ∈ G, ?ξg?∗≤ λwgand ξg
j= 0 if j / ∈ g,
(3)
where ξ = (ξg)g∈Gand ξg
the vector ξgin Rp. Then, problems (2) and (3) are dual to
each other and strong duality holds. In addition, the pair
of primal-dual variables {v,ξ} is optimal if and only if ξ
is a feasible point of the optimization problem (3), and
jdenotes the j-th coordinate of
v = u −?
∀g ∈ G,
g∈Gξg,
v⊤
or v|g= 0.
?
|gξg= ?v|g??ξg?∗and ?ξg?∗= λwg,
4It is easy to show that the dual norm of the ℓ2norm is the ℓ2
norm itself. The dual norm of the ℓ∞is the ℓ1norm.

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Proximal Methods for Sparse Hierarchical Dictionary Learning
For space limitation reasons, we have omitted all the de-
tailed proofs from this section. They will be available in
a longer version of this paper. Note that we focus here on
specific tree-structured groups, but the previous lemma is
valid regardless of the nature of G.
ThestructureofthedualproblemofEq.(3), i.e., thesepara-
bility of the (convex) constraints for each vector ξg, g ∈ G,
makes itpossibletouseblockcoordinateascent(Bertsekas,
1999). Such a procedure is presented in Algorithm 1. It op-
timizes sequentially Eq. (3) with respect to the variable ξg,
while keeping fixed the other variables ξh, for h ?= g. It is
easy to see from Eq. (3) that such an update for a group g
in G amounts to the orthogonal projection of the vector
u|g−?
norm ?.?∗. We denote this projection Π∗
h?=gξh
|gonto the ball of radius λwg of the dual
λwg.
Algorithm 1 Block coordinate ascent in the dual
Inputs: u ∈ Rpand set of groups G.
Outputs: (v,ξ) (primal-dual solutions).
Initialization: v = u, ξ = 0.
while ( maximum number of iterations not reached ) do
for g ∈ G do
v ← u −?
ξg← Π∗
end for
end while
v ← u −?
h?=gξh.
λwg(v|g).
g∈Gξg.
3.3. Convergence in One Pass
In general, Algorithm 1 is not guaranteed to solve exactly
Eq. (2) in a finite number of iterations. However, when ?.?
is the ℓ2or ℓ∞norm, and provided that the groups in G
are appropriately ordered, we now prove that only one pass
of Algorithm 1, i.e., only one iteration over all groups, is
sufficient to obtain the exact solution of Eq. (2). This result
constitutes the main technical contribution of the paper.
Beforestatingthisresult, weneedtointroduceakeylemma
that shows that, given two nested groups g,h such that g ⊆
h ⊆ {1,...,p}, if ξgis updated before ξhin Algorithm 1,
then the optimality condition for ξgis not perturbed by the
update of ξh.
Lemma 2 (Projections with nested groups)
Let ?.? denote either the ℓ2or ℓ∞norm, and g and h be
two nested groups—that is, g ⊆ h ⊆ {1,...,p}. Let v be a
vector in Rp, and let us consider the successive projections
κg △= Π∗
tg(v|g) and κh△= Π∗
th(v|h− κg)
with tg,th> 0. Then, we have as well κg=Π∗
tg(v|g−κh
|g).
The previous lemma establishes the convergence in one
pass of Algorithm 1 in the case where G contains only two
nested groups g ⊆ h, provided that ξgis computed be-
fore ξh. In the following proposition, this lemma is ex-
tended to general tree-structured sets of groups G:
Proposition 1 (Convergence in one pass) Suppose
the groups in G are ordered according to ? and that
the norm ?.? is either the ℓ2or ℓ∞norm5. Then, after
initializing ξ to 0, one pass of Algorithm 1 with the order ?
gives the solution of Eq. (2).
that
Proof sketch.
ceed by induction, by showing that we keep the optimality
conditions of Eq. (3) satisfied after each update in Algo-
rithm 1. The induction is initialized by the leaves. Once
the induction reaches the last group, i.e., after one com-
pletepassover G, thedualvariableξ satisfiestheoptimality
conditions for Eq. (3), which implies that {v,ξ} is optimal.
Since strong duality holds, v is the solution of Eq. (2).
The proof relies on Lemma 2. We pro-
3.4. Efficient Computation of the Proximal Operator
Since one pass of Algorithm 1 involves |G| = p projections
onto the ball of the dual norm (respectively the ℓ2and the
ℓ1norms) of vectors in Rp, a naive implementation leads to
a complexity in O(p2), since each of these projections can
be obtained in O(p) operations (see Mairal et al., 2010, and
references therein). However, the primal solution v, which
is the quantity of interest, can be obtained with a better
complexity, as exposed below:
Proposition 2 (Complexity of the procedure)
i) For the ℓ2norm, the primal solution v of Algorithm 1 can
be obtained in O(p) operations.
ii) For the ℓ∞norm, v can be obtained in O(pd) opera-
tions, where d is the depth of the tree.
The linear complexity in the ℓ2norm case results from a
recursive implementation. It exploits the fact that each pro-
jection amounts to a scaling, whose factor can be found
without explicitly performing the full projection at each it-
eration. As for the ℓ∞norm, since all the groups at a depth
k ∈ {1,...,d} do not overlap, the cost for performing all
the projections at this depth k is O(p), which leads to a
total complexity of O(dp). Note that d could depend on
p as well. For instance, in an unbalanced case, the worse
case could be d = O(p), in a balanced tree, one could have
d = O(log(p)). In practice, the structures we have consid-
ered are relatively flat, with a depth not exceeding d = 5.
Details will be provided in a longer version of this paper.
3.5. Learning the Dictionary
We alternate between the updates of D and A, minimizing
over one while keeping the other variable fixed.
5Interestingly, we have observed that this was not true in gen-
eral when ?.? is an ℓqnorm, for q ?= 2 and q ?= ∞.

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Proximal Methods for Sparse Hierarchical Dictionary Learning
Updating D.
inversion free procedure of Mairal et al. (2010) for up-
dating the dictionary. This method consists in a block-
coordinate scheme over the columns of D. Specifically,
we assume that the domain set D has the form
We have chosen to follow the matrix-
Dµ
△= {D ∈ Rm×p, µ?dj?1+ (1 − µ)?dj?2
2≤ 1}, (4)
or D+
particular domain sets is motivated by the experiments of
Section 4. For natural image patches, the dictionary ele-
ments are usually constrained to be in the unit ℓ2norm ball
(i.e., D = D0), while for topic modeling, the dictionary
elements are distributions of words and therefore belong
to the simplex (i.e., D = D+
1). The update of each dic-
tionary element amounts to performing a Euclidean pro-
jection, which can be computed efficiently (Mairal et al.,
2010). Concerning the stopping criterion, we follow the
strategy from the same authors and go over the columns
of D only a few times, typically 5 in our experiments.
µ
△= Dµ∩Rm×p
+
, with µ ∈ [0,1]. The choice for these
Updating the vectors αi.
the columns of A is built upon the results derived in Sec-
tion 3.2. We have shown that the proximal operator from
Eq. (2) can be computed exactly and efficiently. It makes
it possible to use fast proximal techniques, suited to non-
smooth convex optimization.
The procedure for updating
Specifically, we have tried the accelerated scheme from
both Nesterov (2007) and Beck & Teboulle (2009), and fi-
nally opted for the latter since, for a comparable level of
precision, fewer calls of the proximal operator are required.
The procedure from Beck & Teboulle (2009) basically fol-
lows Section 3.1, except that the proximal operator is not
directly applied on the current estimate, but on an auxiliary
sequence of points that linearly combines past estimates.
This algorithm has an optimal convergence rate in the class
of first-order techniques, and also allows warm restarts,
which is crucial in our alternating scheme. Furthermore,
positivity constraints can be added on the domain of A,
by noticing that for our norm Ω and any u ∈ Rp, adding
these constraints when computing the proximal operator is
equivalent to solving
min
v∈Rp
1
2?[u]+− v?2
2+ λΩ(v),
with ([u]+)j
decompositions to model text corpora in Section 4.
△= max{uj,0}. We will indeed use positive
Finally, we monitor the convergence of the algorithm by
checking the relative decrease in the cost function. We also
investigated the derivation of a duality gap, but this implies
the computation of the dual norm Ω∗for which no closed-
form is available; computing approximations of Ω∗based
on bounds from Jenatton et al. (2009) turned out to be too
slow for our experiments.
4. Experiments
4.1. Natural Image Patches
This experiment studies whether a hierarchical structure
can help dictionaries for denoising natural image patches,
and in which noise regime the potential gain is significant.
We aim at reconstructing corrupted patches from a test set,
after having learned dictionaries on a training set of non-
corrupted patches. Thoughnottypicalinmachinelearning,
this setting is reasonable in the context of images, where
lots of non-corrupted patches are easily available.6
We have extracted 100,000 patches of size m = 8×8 pix-
els from the Berkeley segmentation database of natural im-
ages7, which contains a high variability of scenes. We have
then split this dataset into a training set Xtr, a validation
set Xval, and a test set Xte, respectively of size 50,000,
25,000, and 25,000 patches. All the patches are centered
and normalized to have unit ℓ2norm.
For the first experiment, the dictionary D is learned on Xtr
using the formulation of Eq. (1), with µ = 0 for Dµde-
fined in Eq. (4). The validation and test sets are corrupted
by removing a certain percentage of pixels, the task being
to reconstruct the missing pixels from the known pixels.
We thus introduce for each element x of the validation/test
set, a vector ˜ x, equal to x for the known pixel values and
0 otherwise. In the same way, we define˜D as the matrix
equal to D, except for the rows corresponding to missing
pixel values, which are set to 0. By decomposing ˜ x on˜D,
we obtain a sparse code α, and the estimate of the recon-
structed patch is defined as Dα. Note that this procedure
assumes that we know which pixel is missing and which is
not for every element x.
The parameters of the experiment are the regularization pa-
rameter λtrused during the train step, the regularization
parameter λteused during the validation/test step, and the
structure of the tree. For every reported result, these pa-
rameters have been selected by taking the ones offering
the best performance on the validation set, before report-
ing any result from the test set. The values for the reg-
ularization parameters λtr,λtewere tested on a logarith-
mic scale {2−10,2−9,...,22}, and then further refined on
a finer logarithmic scale of factor 2−1/4. For simplicity
reasons, we have chosen arbitrarily to use the ℓ∞-norm
in the structured norm Ω, with all the weights equal to
one. We have tested 21 balanced tree structures of depth
3 and 4, with different branching factors p1,p2,...,pd−1,
where d is the depth of the tree and pk, k ∈ {1,...,d − 1}
6Note that we study the ability of the model to reconstruct
independent patches, and additional work is required to apply our
framework to a full image processing task, where patches usually
overlap (Elad & Aharon, 2006).
7www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/