arXiv:1709.03688v1 [] 12 Sep 2017

Joint Dictionaries for Zero-Shot Learning
Soheil Kolouri ∗

Mohammad Rostami ∗

Yuri Owechko

Kyungnam Kim

HRL Laboratories, LLC

University of Pennsylvania

HRL Laboratories, LLC

HRL Laboratories, LLC

skolouri@hrl.com

mrostami@seas.upenn.edu

yowechko@hrl.com

kkim@hrl.com

Abstract
A classic approach toward zero-shot learning (ZSL) is to
map the input domain to a set of semantically meaningful attributes that could be used later on to classify unseen classes
of data (e.g. visual data). In this paper, we propose to learn
a visual feature dictionary that has semantically meaningful
atoms. Such dictionary is learned via joint dictionary learning
for the visual domain and the attribute domain, while enforcing the same sparse coding for both dictionaries. Our novel attribute aware formulation provides an algorithmic solution to
the domain shift/hubness problem in ZSL. Upon learning the
joint dictionaries, images from unseen classes can be mapped
into the attribute space by finding the attribute aware joint
sparse representation using solely the visual data. We demonstrate that our approach provides superior or comparable performance to that of the state of the art on benchmark datasets.

Introduction
Most classification algorithms require a large pool of manually labeled data to learn the optimal parameters of a classifier. The recent exponential growth of visual data, the growing need for fine-grained multi-label annotations, and consistent emergence of new classes (e.g. new products), however, has rendered manual labeling of data practically infeasible. Transfer learning has been proposed as a remedy
to deal with this issue (Lampert, Nickisch, and Harmeling
2009). The idea is to learn on a limited number of classes
and then through knowledge transfer, learn how to classify
images from the new classes either using only few labeled
data points, i.e. few- and one-shot learning (Fei-Fei, Fergus,
and Perona 2006), or in the extreme case without any labeled data, i.e. zero-shot learning (ZSL) (Lampert, Nickisch,
and Harmeling 2009). These transfer learning approaches
address the challenge of annotated data unavailability and
open the door towards lifelong learning machines.
To learn target classes with no labeled data, one needs
to be able to generalize the relationship between the source
data and its labels to the target classes. To address this challenge in ZSL, an intermediate shared space (i.e. the space
of semantic attributes) is exploited, which allows for knowledge transfer from labeled classes to the unlabeled classes.
∗

Equal contribution
Copyright c 2018, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.

The overarching idea in ZSL is that the source and the target classes share common attributes. The semantic attributes
(e.g., can fly, is green) are often provided as accessible
side information (e.g. verbal description of a class), which
uniquely describe classes of data. To achieve ZSL the relationship between seen data and its corresponding attributes
are first learned in the training phase. In testing stage, this
allows for parsing a target image from an unseen class into
its semantic attributes to predict corresponding label.
To clarify the ZSL core idea and the required steps to
perform ZSL, consider the following sentence: ‘Tardigrades
(also known as water bears or moss piglets) are waterdwelling, eight-legged, segmented micro animals’1 . Given
this textual description, one can easily identify the creature in Figure 1, Left as a Tardigrade even though she may
have never seen one before. Performing this task requires
three capabilities: 1) parsing the textual information into semantic features, so we can describe the class Tardigrade
as ‘bear-like’, ‘piglet-like’, ‘water-dwelling’, ‘eight-legged’,
‘segmented’, and ‘microscopic animal’, 2) parsing the image into its visual attributes (See Figure 1), and 3) matching the parsed visual features to the parsed textual information which often requires extensive prior knowledge. Recent
textual features extracted from large unlabeled text corpora;
including word2vec (Mikolov et al. 2013) and glove (Pennington, Socher, and Manning 2014) enable a learner to efficiently parse textual information. Deep convolutional neural
networks (CNNs) (Krizhevsky, Sutskever, and Hinton 2012;
Simonyan and Zisserman 2014; He et al. 2016; Huang et al.
2017) have revolutionized the field of computer vision and
they enable a learner to extract rich visual features from images. An extensive body of work in the field of ZSL is concentrated on modeling the relationship between visual features and semantic attributes (Palatucci et al. 2009; Akata et
al. 2013; Socher et al. 2013; Norouzi et al. 2014; Lampert,
Nickisch, and Harmeling 2009; Zhang and Saligrama 2015;
Ding, Shao, and Fu 2017).
In this paper, we provide a novel approach to model the
relationship between the visual features and the textual information. Our specific contributions are:
1. New formulation of ZSL via joint dictionary learning
2. Extending the classic joint dictionary learning formula1

Source: Wikipedia

Figure 1: High-level overview of our approach. Left & right:
visual and attribute feature extraction and representation using union of subspaces. Middle: constraining the dictionary
atoms to be coupled.
tion to an attribute aware formulation that addresses the
domain shift/adaptation problem (Kodirov et al. 2015)
3. Demonstrating the benefit of a transductive learning
scheme to reduce the hubness phenomenon (Dinu, Lazaridou, and Baroni 2014; Shigeto et al. 2015)

Related Work
ZSL methods often focus on learning the relationship between the visual space and the semantic attribute space.
Palatucci et al. (Palatucci et al. 2009) proposed to learn a
linear compatibility between the visual space and the semantic attribute space. Lampert et al. (Lampert, Nickisch, and
Harmeling 2009) posed the problem as an attribute classification problem and learned individual binary attribute classifiers in the training stage and used the ensemble of classifiers to map visual features to their semantic attributes. Yu
and Aloimonos (Yu and Aloimonos 2010) approached the
problem from a probabilistic point of view and proposed to
use generative models to learn prior distributions for image
features with respect to each attribute. More recently, various authors have proposed to embed image features and semantic attributes in a shared metric space (i.e. a latent embedding) (Akata et al. 2013; Romera-Paredes and Torr 2015;
Zhang and Saligrama 2015) while forcing the embedded
representations for image features and their corresponding
semantic attributes to be similar. Akata et al. (Akata et al.
2013), for instance, proposed a model that embeds the image features and the semantic attributes in a common space
(i.e. a latent embedding) where the compatibility between
them is measured via a bilinear function. Similarly, RomeraParedes and Torr (Romera-Paredes and Torr 2015) utilized a
principled choice of regularizers that enable the authors to
derive a simple closed form solution to learn a linear mapping that embeds the image features and the semantic attributes in a low dimensional shared linear subspace. Others
have identified the major problems and challenges in ZSL
to be the domain shift problem (Kodirov et al. 2015) and
the hubness phenomena (Dinu, Lazaridou, and Baroni 2014;
Shigeto et al. 2015). In short, the domain shift problem

raises from the fact that the distribution of features corresponding to the same attribute for seen and unseen images could be very different (e.g. stripes of tigers versus
zebras). The hubness problem, on the other hand, states
that there will often be attributes that are similar (have
small distance) to vastly different visual features in the embedding space. Various transductive approaches are presented to overcome the hubness problem (Fu et al. 2015;
Yu et al. 2017).
The use of sparse dictionaries to model the space of visual
features and semantic attributes as union of linear subspaces
has been shown to be an effective modeling scheme in recent
ZSL papers (Yu et al. 2017; Isele, Rostami, and Eaton 2016;
Kodirov et al. 2015; Zhang and Saligrama 2015). Zhang
et al. (Zhang and Saligrama 2015) showed that modeling
the test image features as sparse linear combination of train
image features is beneficial and formulated a ZSL method
based on this principal. Using similar ideas, Isele et. al.
(Isele, Rostami, and Eaton 2016) used joint dictionary learning to learn a dynamical control system using high level
task descriptors in an online lifelong zero-shot reinforcement learning setting. Our JD-ZSL build on similar ideas
as in (Yu et al. 2017; Isele, Rostami, and Eaton 2016;
Kodirov et al. 2015) and introduce a novel ZSL method
based on learning joint sparse dictionaries for the image
features and the semantic attributes. At its core, JD-ZSL
is equipped with a novel entropy minimization regularizer
(Grandvalet and Bengio 2004), which facilitates the solution
to the ZSL problem by reducing the domain shift effect. We
further show that a transductive approach applied to our attribute aware JD-ZSL formulation provide state-of-the-art or
close to state-of-the-art performance on various benchmark
datasets. Finally it should be noted that the idea of using
joint dictionaries to map data from a given metric space to a
second related space was pioneered by Yang et al. (Yang et
al. 2010) in super-resolution applications.
Figure 1 captures the gist of our idea. Visual features
are extracted via CNNs, left sub-figure, and the semantic
attributes are provided via textual feature extractors like
word2vec or via human annotations, right sub-figure. Both
the visual features and the semantic attributes are assumed
to be representable sparsely in a shared union of linear subspaces, left and right sub-figures. The idea here is then to
enforce the sparse representation vectors for both domains
be equal and thus effectively couple the learned dictionaries for the the visual and the attribute spaces. The intuition
from a co-view perspective (Yu et al. 2014) is that both the
visual and the attribute features provide information about
the same class, and so each can augment the learning of the
other. Each underlying class is common to both views, and
one can find task embeddings that are consistent for both
the visual features and their corresponding attributes. Having learned the coupled dictionaries, zero-shot classification
can be performed by mapping images of unseen classes into
the attribute space, where classification can be simply done
via nearest neighbor or via a more elaborate scheme like
label propagation. Given the coupled nature of the learned
dictionaries, an image could be mapped to its semantic attributes by first finding the sparse representation with respect

to the visual dictionary, and next the semantic attribute dictionary can be used to recover the attribute vector from the
joint sparse representation which could then be used for classification.

Problem Statement and Technical Rational
Consider a visual feature metric space X of dimension p, an
attribute metric space Z with dimension q, and a class label
set Y with dimension K which ranges over a finite alphabet
of size K (images can potentially have multiple memberships to the classes). As an example X = Rp for the visual features extracted from a deep CNN and Z = {0, 1}q
when a binary code of length q is used to identify the presence/absence of various characteristics in an object (Lampert, Nickisch, and Harmeling 2009). We are given a labeled dataset D = {((xi ; zi ), yi )}N
i=1 of features of seen
images and their corresponding semantic attributes, where
∀i : xi ∈ X , zi ∈ Z, and yi ∈ Y. We are also given the
unlabeled attributes of unseen classes D0 = {z0j }M
j=1 (i.e. we
have access to textual information for a wide variety of objects but do not have access to the corresponding visual information). In ZSL the set of seen and unseen classes are disjoint and it is assumed that the semantic attributes are class
specific. The goal is then to use D and D0 to learn the relationship between X and Z so when an unseen image (image
from an unseen class) is fed to the system, its corresponding
attributes and consequently its label could be predicted. Finally, we assume that ψ : Z → Y is the mapping between
the attribute space and the label space and ψ is a known linear mapping, y = ψ(z) = Vz.
To further clarify the problem, consider an instance of
ZSL in which features extracted from images of horses and
tigers are included in seen visual features X = [x1 , ..., xN ],
where xi ∈ X , but X does not contain features from images
containing zebras. On the other hand, the semantic attributes
contain information of all seen Z = [z1 , ..., zN ] for zi ∈ Z
and unseen Z 0 = [z01 , ..., z0M ] for z0j ∈ Z classes including
the zebras. Intuitively, by learning the relationship between
the image features and the attributes “has hooves”, “has
mane”, and “has stripes” from the seen images, we must be
able to assign an image of a zebra to its corresponding attribute, while we have never seen a zebra before. More formally, we want to learn the mapping φ : X → Z which relates the visual space and the attribute space. Having learned
this mapping, for an unseen image one can recover the corresponding attribute vector using the image features and then
classify the image using the mapping y = (ψ ◦ φ)(x), where
‘◦’ represents function composition.

Technical Rational
For the rest of our discussion we assume that X = Rp ,
Z = Rq , and Y = RK . The simplest ZSL approach is to assume that the mapping φ : Rp → Rq is linear, φ(x) = W T x
p×q
where
then minimize the regression error
P W T∈ R , and
1
2
kW
x
−
z
k
to
learn W . Despite existence of a
i
i
2
i
N
closed form solution for W , the solution
P contains the inverse
of the covariance matrix of X, ( N1 i (xi xTi ))−1 , which requires a large number of data points for accurate estimation.

To overcome this problem, various regularizations are considered for W . Decomposition of W as W = P ΛQ, where
P ∈ Rp×l , Λ ∈ Rl×l , Q ∈ Rl×q , and l < min(p, q) can
also be helpful. Intuitively, P is a right linear operator that
projects x’s into a shared low dimensional subspace, Q is
a left linear operator that projects z into the same shared
subspace, and Λ provides a bi-linear similarity measure in
the shared subspace. The regression
P problem then can be
transformed into maximizing N1 i xTi P ΛQzi , which is a
weighted correlation between the embedded x’s and z’s.
This is the essence of many ZSL techniques including Akata
et al. (Akata et al. 2013) and Romera-Paredes et al.(RomeraParedes and Torr 2015). This technique can be extended
to nonlinear mappings using kernel methods. However, the
choice of kernels remains a challenge.
On the other side of the spectrum, the mapping φ : Rp →
q
R can be chosen to be highly nonlinear, as in deep neural
networks (DNN). Let a DNN be denoted as φ(.; θ), where
θ represents the parameters of the network (i.e. synaptic
weights and
Pbiases). ZSL can then be addressed by minimizing N1 i kφ(xi ; θ) − zi k22 with respect to θ. Alternatively, one can nonlinearly embed x’s and z’s in a shared
metric space via deep nets, f (x; θx ) : Rp → Rl and
g(z; θz ) : Rq → Rl , and maximize
P their similarity measure in the embedded space, N1 i f (xi ; θx )T g(zi ; θz ), as
in (Lei Ba et al. 2015). Nonlinear methods are computationally expensive, require a large training dataset, and can
easily overfit to the training data. On the other hand, linear
ZSL algorithms are efficient, easy to train, and generalizable
but they are often outperformed by nonlinear methods. As a
compromise, we model nonlinearities in data distributions
as union of linear subspaces with coupled dictionaries. By
jointly learning the visual and attribute dictionaries, we effectively model the relationship between the metric spaces.
This allows a nonlinear scheme with a computational complexity comparable to linear techniques.

Zero Shot Learning using Joint Dictionaries
Joint dictionary learning has been proposed to couple related features from two metric spaces (Yang et al. 2010;
Shekhar et al. 2014). Yang et al. (Yang et al. 2010) proposed the approach to tackle the problem of image superresolution and Shekhar et al. (Shekhar et al. 2014) used joint
dictionary learning for multimodal biometrics recognition.
Following a similar framework, the gist of our approach is
to learn the mapping φ : Rp → Rq through two dictionaries, Dx ∈ Rp×r and Dz ∈ Rq×r that model X and [Z, Z 0 ],
respectively, where r > max(p, q). The goal is to find a
shared sparse representation (i.e. sparse code) ai for xi and
zi , such that xi = Dx ai and zi = Dz ai . Below we describe
the training and testing phases of our proposed method.

Training phase
The standard dictionary learning is based on minimizing the
empirical average estimation error N1 kX − Dx Ak2F on a
given training set X, where `1 regularization on A enforces

Zero-Shot Prediction of Unseen Attributes

sparsity:
Dx∗ , A∗ = argmin
Dx ,A

1
kX − Dx Ak2F + λkAk1
N

(1)

s.t. kDx[i] k22 ≤ 1.
Here λ is the regularization parameter, which controls the
[i]
sparsity of A, and Dx is the i’th column of Dx . Alternatively, following the Lagrange multiplier technique, the
Frobenius norm of Dx could be used as a regularizer in place
of the costraint.
In our joint dictionary learning framework, we aim to
learn Dx and Dz such that they share the sparse coefficients
A to represent the seen visual features X and their corresponding attributes Z, respectively. An important twist here
is that the attribute dictionary, Dz , is also required to sparsify the semantic attributes of other (unseen) classes, Z 0 . To
obtain such coupled dictionaries we propose the following
optimization,
1
pλ
(kX − Dx Ak2F +
kAk1 )+
r
Dx ,A,Dz ,B N p
1
1
kZ − Dz Ak2F +
(kZ 0 − Dz Bk2F + (2)
Nq
Mq
qλ
kBk1 )} s.t.:kDx[i] k22 ≤ 1, kDz[i] k22 ≤ 1
r
The above formulation combines the dictionary learning
problem for X and Z by coupling them via A, and also enforces Dz to be a sparsifying dictionary (i.e. a good model)
for Z 0 . The optimization in Eq (2), while convex in each individual term, is highly nonconvex in all variables. Following the approach proposed in (Yang et al. 2012) we use an
Expectation Maximization (EM) like alternation to update
dictionaries Dx and Dz . To do so, we rewrite the optimization problem into the following two steps:
1. For a fixed Dx update Dz via the following optimization:
argmin {

min

1

Dz ,B M q

(kZ 0 − Dz Bk2F +

qλ
kBk1 )+
r

1
kZ − Dz A∗ k2F
Nq
1
λ
s.t. A∗ = argmin kX − Dx Ak2F + kAk1 ,
p
r
A

(3)

kDz[i] k22 ≤ 1
A is found using a Lasso optimization problem, and
FISTA (Beck and Teboulle 2009) is used to update Dz
and B.
2. For a fixed Dz update Dx via:
min kX − Dx A∗ k2F
Dx

1
λ
s.t. A∗ = argmin kZ − Dz Ak2F + kAk1 ,
q
r
A

(4)

kDx[i] k22 ≤ 1,
which involves a Lasso optimization together with a simple regression with a close form solution.

In the testing phase we are only given the extracted features
from unseen images, X 0 = [x01 , ..., x0l ] ∈ Rp×l and the goal
is to predict their corresponding semantic attributes. Here
we introduce a progression of methods, which clarifies the
logic behind our method, and enables us to efficiently predict
the semantic attributes of the unseen images based on the
learned dictionaries in the training phase.
Attribute Agnostic Prediction The attribute agnostic
(AAg) formulation, is the naive way of predicting semantic
attributes from an unseen image x0i . In the AAg formulation,
we first find the sparse representation αi of the unseen image x0i with respect to the learned dictionary Dx by solving
the following Lasso problem,
1
λ
αi = argmina kxi − Dx ak22 + kak1 .
p
r

(5)

Here, one can simply use αi and compare it to the sparse
codes of the unseen attributes, bj . In our experiments, however, we found that this approach is not suitable in our JDZSL setting as the dictionaries could have redundant atoms
that cause two similar image features or attributes to have
different sparse codes. Instead, we do the comparison in the
attribute space and predict the corresponding attribute via
ẑi = Dz αi . In the attribute-agnostic formulation, the sparse
coefficients are calculated without any information from the
attribute space. Not using the information from the attribute
space would lead to the domain shift problem, in the sense
that there is no guarantee that αi would reconstruct a meaningful attribute in Z. In other words, ẑi = Dz αi could be
far from the unseen attributes, z0m , and therefore could not
be assigned to any known attribute with high confidence. To
alleviate this problem we progress to an extended solution,
which we denote as the Attribute Aware (AAw) prediction.
Attribute Aware Prediction In the attribute-aware (AAw)
formulation we would like to find the sparse representation
αi to not only approximate the input visual feature, x0i ≈
Dx αi , but also provide an attribute prediction, ẑi = Dz αi ,
that is well resolved in the attribute space and does not suffer
from the domain shift problem. Note that, ideally ẑi = z0m
for some m ∈ {1, ..., M }. To achieve this we define the soft
assignment of ẑi to z0m , denoted by pm , using the Student’s
t-distribution as a kernel to measure similarity between ẑi =
Dz αi and z0m ,
kDz αi −z0m k22 − ρ+1
) 2
ρ
0
kDz αi −zk k22 − ρ+1
) 2
ρ

(1 +
pm (αi ) = P
k (1 +

(6)

where ρ is the kernel parameter. The choice of t-distribution
is due to its long tail and low sensitivity to the choice of
kernel parameter, ρ. Ideally, pm (αi ) = 1 for some m ∈
{1, ..., M } and pj (αi ) = 0 for j 6= m. The ideal softassignment p = [p1 , p2 , ..., pM ] then would be one-sparse
and therefore would have minimum entropy. This motivates
our attribute-aware formulation, which regularizes the AAg

formulation in Equation 5 with the entropy of p.
X
1
αi = argmina kx0i − Dx ak22 − γ
pm (a)log(pm (a))
p
m
|
{z
}
g(a)

λ
+ kak1
r

(7)

where γ is the regularization parameter for entropy of the
soft-assignment probability vector p. Such entropy minimization scheme has been successfully used in several work
(Grandvalet and Bengio 2004; Huang, Tran, and Tran 2016)
whether as a sparsifying regularization or to boost the confidence of classifiers. We note that the entropy regularization
enforces the prediction to be close to one of the unseen attributes, but it can potentially backfire in that a low-entropy
solution (aligned to a prototype) doesn’t necessarily have to
be the correct solution. In our experiments, we consistently
observed higher performance for the AAw formulation.
The entropy regularization turns the optimization in Eq.
(7) into a nonconvex problem. In (Huang, Tran, and Tran
2016), the authors use a generalized gradient descent approach similar to FISTA to optimize this non-convex problem. We use a similar scheme to optimize the objective function in Eq. (7). In short, we relax g(a) using its quadratic approximation around the previous estimation of a, ak−1 , and
update a as the solution of the following problem
1
ka − (ak−1 − t∇g(ak−1 ))k22 +
2t
(8)
λ
kak1
r
Equation (8) is a LASSO problem and can be solved efficiently using FISTA. It only remains to compute ∇g:
ak = argmina

1
∇g(a) = DxT (Dx a − x0 )−
p
X
γ
P
{(1 + log(pm (a)))×
2
( k lk (a)) m
X ∂lk (a)
∂lm (a) X
(
lk (a) − lm (a)
)}
∂a
∂a
k

(9)

k

where:
kDz a − z0m k22 − ρ+1
) 2 ,
ρ
∂lm (a)
kDz a − z0m k22 − ρ+3
ρ+1 T
=−
(Dz (Dz a − z))(1 +
) 2 .
∂a
ρ
ρ

lm (a) = (1 +

Due to the non-convex nature of the objective function, a
good initialization is needed to achieve a sensible solution.
Therefore we initialize α from the solution of the AAg formulation. Finally the corresponding attributes are estimated
by ẑi = Dz αi , for i = 1, ..., l.

From Predicted Attributes to Labels
In order to predict the image labels, one needs to assign the
predicted attributes, Ẑ = [ẑ1 , ..., ẑl ], to the M attributes of

the unseen classes Z 0 (i.e. prototypes). This task can be performed in two ways, namely the inductive approach and the
transductive approach. In the inductive scheme the inference
could be performed using a nearest neighbor (NN) approach
in which label of each individual ẑi is assigned to be the label of its nearest neighbor z0m . In such approach the structure
of ẑi ’s is not taken into account and the hubness problem
could easily degrade the performance of the ZSL algorithm.
Looking at the t-SNE embedding visualization (Maaten and
Hinton 2008) of ẑi ’s and z0m ’s in Figure 2, details are explained later, it can be seen that NN does not provide an
optimal label assignment.
In the transductive setting, on the other hand, the attributes
for all test images (i.e. unseen) are first predicted to form
Ẑ = [ẑ1 , ..., ẑl ]. Next, a graph is formed on [Z 0 , Ẑ] where
the labels for Z 0 are known and the task is to infer the
labels of Ẑ. This problem can be formulated as a graphbased semi-supervised label propagation (Belkin, Matveeva,
and Niyogi 2004; Zhou et al. 2003). We follow the work
of Zhou et al. (Zhou et al. 2003) and spread the labels of
Z 0 to Ẑ. More precisely, we first reduce the dimension of
[Z 0 , Ẑ] via t-SNE (Maaten and Hinton 2008) and then form
a graph in the lower dimension and perform label propagation on this graph. Figure 2 reconfirms that label propagation in a transductive setting could significantly improve
the performance of ZSL and resolve the hubness and domain shift issues as also demonstrated in (Fu et al. 2015;
Yu et al. 2017).

Theoretical Discussion
The core step for ZSL in our scheme is to compute the joint
sparse representation for an unseen image. Note that in the
testing phase, the sparse representation a is estimated using
(5), while the dictionaries are learned for optimal sparse representations as in (2). More specifically, we need to demonstrate that the following two problems lead to close approximations:
α∗ = argmina kx − Dx ak22 + kz − Dz ak22 + λkak1
   
x
Dx
= argmina k
−
ak22 + λkak1
z
Dz
α+ = argmina kx − Dx ak22 + λkak1 ,
(10)
in order to conclude that we can solve for α+ in ZSL
regime (i.e. prediction attributes for unseen images) to estimate α∗ with good accuracy. Note that the major challenge in the testing phase is that we are using the dictionary Dx ∈ Rp×r to find the shared sparse parameters, α,
instead of D̃ = [Dx , Dz ]T ∈ R(p+q)×r . To study the effect
of this change, we first point out that Eq. 1 can be interpreted as result of a maximum a posteriori (MAP) inference
from a Bayesian perspective. This means that from a probabilistic perspective, α’s are drawn from a Laplacian distribution and the dictionary D is a Gaussian matrix with elements drawn i.i.d: dij ∼ N (0, σ 2 ). This means that given
a drawn dataset, we learn MAP estimate of a Gaussian matrix. In order to analyze the effect, we rely on the following

Figure 2: Attributes predicted from the input visual features for the unseen classes of images for AWA dataset using our attribute
agnostic and attribute aware formulations respectively in top and bottom rows. The nearest neighbor and label propagation
assignment of the labels together with the ground truth labels are visualized. It can be seen that the attribute aware formulation
together with the label propagation scheme overcomes the hubness and domain shift problems. Best seen in color.
theorem about LASSO with Gausian matrices (Negahban et
al. 2009):
Theorem 1 (Negahban et al. 2009): Let αs be the unique
sparse solution of the linear system x = Da with kak0 = k
and D ∈ Rp×n . If α‡ is the LASSO solution for the system from noisy observations,
then with high probability:
q
kαs − α‡ k2 ≤ c0 k logp r , where c0 ∈ R+ is a constant
which depends on the loss function which measures the data
fidelity, here the Euclidean distance.
Lemma 1: Attribute prediction error in ZSL setting is
1
upper-bounded proportional to ( √1p + √q+p
).
Proof: note that if α∗ is a solution of [xT , zT ]T = D̃a,
trivially it is also a solution for x = Dx a as well. Now using
Theorem 1:
kz∗ − z+ k ≤ kDx (α∗ − α+ )k
p
1
1
)
kDx (α∗ − α+ )k ≤ c0 kDz k2 k log r( √ + √
p
q+p
(11)
Note we have used the triangular inequality first and then the
theorem in the above deduction and k · k2 denotes spectral
norm for a matrix. This result accords with intuition. First,
it advises sparseness of z, i.e. smaller k, decreases the error
which means that a good sparsifying dictionary would lead
to less ZSL error.
the error is proportional to in√
√ Second,
verse of both p and p + q, meaning that rich visual and
attribute descriptions lead to minimal ZSL error. This suggests that for our approach to work, existence of a good spar-

sifying dictionary as well as rich visual and attribute data is
essential. Finally, although increasing the number of dictionary columns r intuitively can improve sparsity, i.e. decrease
k, this result shows that it can potentially increase the ZSL
error, and should be tuned for an optimal performance.

Experiments
We carried out experiments on three benchmark ZSL
datasets and empirically evaluated the resulting performance
against nascent ZSL algorithms.
Datasets: We conducted our experiments on three benchmark datasets namely: the Animals with Attributes (AwA1)
(Lampert, Nickisch, and Harmeling 2014), the SUN attribute (Patterson and Hays 2012), and the Caltech-UCSDBirds 200-2011 (CUB) bird (Wah et al. 2011) datasets. The
AwA1 dataset is a coarse-grained dataset containing 30475
images of 50 types of animals with 85 corresponding attributes for these classes. Semantic attributes for this dataset
are obtained via human annotations. The images for the
AWA1 dataset are not publicly available; therefore we use
the publicly available features of dimension 4096 extracted
from a VGG19 convolutional neural network, which was
pretrained on the ImageNet dataset. Following the conventional usage of this dataset, 40 classes are used as source
classes to learn the model and the remaining 10 classes are
used as target (unseen) classes to test the performance of
zero-shot classification. The SUN dataset is a fine-grained
dataset and contains 717 classes of different scene categories
with 20 images per category (14340 images total). Each im-

Method
(Romera-Paredes and Torr 2015)‡
(Zhang and Saligrama 2015)†
(Zhang and Saligrama 2016)†
(Bucher, Herbin, and Jurie 2016)†
(Xu et al. 2017)†
(Li et al. 2017) †
(Ye and Guo 2017)†
(Ding, Shao, and Fu 2017)†
(Wang and Chen 2017)†
(Kodirov, Xiang, and Gong 2017)†
Ours
AAg (5)
Ours
AAw (6)
Ours Transductive AAw (TAAw)
Ours
TAAw hit@3
Ours
TAAw hit@5

SUN
82.10
82.5
82.83
84.41
83.5
85.40
86.0
91.0
82.05
83.22
85.90
94.52
98.15

CUB
30.41
42.11
43.29
53.6
61.79
57.14
45.2
42.7
61.4
35.81
38.36
47.12
58.19
69.67

AwA
75.32
76.33
80.46
77.32
84.5
87.22
85.66
82.8
79.8
84.7
77.73
83.33
88.23
91.73
97.13

Table 1: Zero-shot classification results for three benchmark
datasets. All methods use VGG19 features trained on the ImageNet dataset and the original continuous (or binned) attributes provided by the datasets. Here, † indicates that the
results are extracted directly from the corresponding paper,
‡ indicates that the results are reimplemented with VGG19
features, and − indicates that the results are not reported.

age is annotated with 102 attributes that describe the corresponding scene. Following (Lampert, Nickisch, and Harmeling 2014), 707 classes are used to learn the dictionaries and
the remaining 10 classes are used for testing. The CUB200
dataset is a fine-grained dataset containing 200 classes of
different types of birds with 11788 images with 312 attributes and boundary segmentation for each image. The attributes are obtained via human annotation. The dataset is
divided into four almost equal folds, where three folds are
used to learn the model and the fourth fold is used for testing.
For both SUN and CUB200-2011 datasets we used features
from VGG19 trained on the ImageNet dataset, which have
4096 dimensions. We note that our results using ResNet50
and DenseNet (Huang et al. 2017) features will be published
in an extended version of this paper.
Tuning parameters: The optimization regularization parameters λ, ρ, γ as well as the number of dictionary atoms
r need to be tuned for maximal performance. We used standard k-fold cross validation to search for the optimal parameters for each dataset. After splitting the datasets accordingly
into training, validation, and testing sets, we used performance on the validation set for tuning the parameters in a
brute-force search. we used the common evaluation metrics
in ZSL, flat hit@K classification accuracy, to measure the
performance. This means that a test image is said to be classified correctly if it is classified among the top K predicted
labels. We report hit@1 rate to measure ZSL image classification performance and hit@3 and hit@5 for image retrieval
performance. Each experiment is performed ten times and
the mean is reported in Tabel 1.
Results: Figure 2 demonstrates the 2D t-SNE embedding for predicted attributes and actual class attributes of
the AWA dataset. The actual attributes are depicted by col-

ored circles with black edges. The first column of Figure
2 demonstrates the attribute prediction for AAg and AAw
formulations. It can be clearly seen that the entropy regularization in AAw formulation improves the clustering quality,
decreases data overlap, and reduces the domain shift problem. The nearest neighbor label assignment is shown in the
second column, which demonstrates the domain shift and
hubness problems with NN label assignment in the attribute
space. The third column of Figure 2 shows the transductive
approach in which a label propagation is performed on the
graph of the predicted attributes. Note that the label propagation addresses the domain shift and hubness problem and
when used with the AAw formulation provides significantly
better zero-shot classification accuracy.
Performance comparison results are summarized in Table
1. As pointed out by Xian et al. (Xian et al. 2017) the variety
of used image features (e.g. various DNNs and various combinations of these features) as well as the variation of used
attributes (e.g. word2vec, human annotation), and different
data splits make direct comparison with the ZSL methods in
the literature very challenging. In Table 1 we provide a fair
comparison of our JDZSL performance to the recent methods in the literature. All compared methods use the same
visual features (i.e. VGG19) and the same attributes (i.e.
the continuous or binned) provided in the dataset. Table 1
provides a comprehensive explanation of the shown results.
Note that our method achieves state-of-the-art or close to
state-of-the-art performance.
We report the hit@1 accuracy on unseen classes in the
first nine rows of the table to measure image classification
performance. For the sake of transparency and to provide
the complete picture to the reader, we included results for
the AAg formulation using nearest neighbor, the AAw using nearest neighbor, and AAw using the transductive approach, denoted as transductive attribute aware (TAA) formulation. As it can be seen, while the AAw formulation
significantly improves the AAg formulation and adding the
transductive approach (i.e. label propagation on predicted attributes) to the AAw formulation further boosts the classification accuracy, as also shown in Figure 2. In addition, our
approach leads to better and comparable performance in all
three datasets which include zero-shot scene and object recgonition tasks. More importantly, while the other methods
can perform well on a specific dataset, our algorithm leads
to competitive performance on all the three datasets.

Conclusions
A ZSL formulation is developed that models the relationship between visual features and semantic attributes via joint
sparse dictionaries. We showed that while a classic joint
dictionary learning approach suffers from the domain shift
problem, an entropy regularization scheme can help with
this phenomenon and provide superior performance. In addition, we demonstrated that a transductive approach towards
assigning labels to the predicted attributes can boost the
performance considerably and lead to state-of-the-art zeroshot classification. Finally, we compared our method to the
nascent approaches in the literature and demonstrated its
competitiveness on benchmark datasets.

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