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Compositional Dictionaries for Domain Adaptive Face Recognition

We present a dictionary learning approach to compensate for the transformation of faces due to changes in view point, illumination, resolution, etc. The key idea of our approach is to force domain-invariant sparse coding, i.e., design a consistent sparse representation of the same face in different domains. In this way, classifiers trained on the sparse codes in the source domain consisting of frontal faces for example can be applied to the target domain (consisting of faces in different poses, illumination conditions, etc) without much loss in recognition accuracy. The approach is to first learn a domain base dictionary, and then describe each domain shift (identity, pose, illumination) using a sparse representation over the base dictionary. The dictionary adapted to each domain is expressed as sparse linear combinations of the base dictionary. In the context of face recognition, with the proposed compositional dictionary approach, a face image can be decomposed into sparse representations for a given subject, pose and illumination respectively. This approach has three advantages: first, the extracted sparse representation for a subject is consistent across domains and enables pose and illumination insensitive face recognition. Second, sparse representations for pose and illumination can subsequently be used to estimate the pose and illumination condition of a face image. Finally, by composing sparse representations for subject and the different domains, we can also perform pose alignment and illumination normalization. Extensive experiments using two public face datasets are presented to demonstrate the effectiveness of our approach for face recognition.


I. INTRODUCTION
Many image recognition algorithms often fail while experiencing a significant visual domain shift, as they expect the test data to share the same underlying distribution as the training data. A visual domain shift is common and natural in the context of face recognition. Such domain shift is due to changes in poses, illumination, resolution, etc.. Domain adaptation [1] is a promising methodology for handling the domain shift by utilizing knowledge in the source domain for problems in a different but related target domain. [2] is one of the earliest works on semi-supervised domain adaptation, where they model data with three underlying distributions: source domain data distribution, target domain data distribution and a distribution of data that is common to both domains. [3] follows a similar model in handling view point changes in the context of activity recognition, where they assume some activities are observed in both source and target Q. Qiu  domains, while some other activities are only in one of the domains. Under the above assumption, certain hyperplanebased features trained in the source domain are adapted to the target domain for improved classification. Domain adaptation for object recognition is studied in [4], where the subspaces of the source domain, the target domain and the potential intermediate domains are modeled as points on the Grassmann manifold. The shift between domains is learned by exploiting the geometry of the underlying manifolds. A good survey on domain adaptation can be found in [4].
Face recognition across domain, e.g., pose and illumination, has proved to be a challenging problem [5], [6], [7]. In [5], the eigen light-field (ELF) algorithm is presented for face recognition across pose and illumination. This algorithm operates by estimating the eigen light field or the plenoptic function of the subject's head using all the pixels of various images. In [8], [6], face recognition across pose is performed using stereo matching distance (SMD). The cost to match a probe image to a gallery image is used to evaluate the similarity of the two images. Both ELF and SMD methods are state-of-the-art methods for face recognition across pose and/or illumination variations. Our proposed domain adaptive dictionary learning approach shows comparable performance to these two methods for face recognition across domain shifts due to pose and illumination variations. In addition, our approach can also be used to estimate the pose and lighting condition of a face, and to perform pose alignment and illumination normalization.
The approach presented here shares some of the attributes of the Tensorfaces method proposed in [7], [9], [10], but significantly differs in many aspects. In the Tensorfaces method, face images observed in different domains, i.e., faces imaged in different poses under different illuminations, form a face tensor. Then a multilinear analysis is performed on the face tensor using the N -mode SVD decomposition to obtain a core tensor and multiple mode matrices, each for a different domain aspect. The N -mode SVD decomposition is similar to the proposed multilinear sparse decomposition shown in Fig. 1, where a given unknown image is decomposed into multiple sparse representations for the given subject, pose and illumination respectively. However, we show through experiments that our method based on sparse decomposition significantly outperforms the N -mode SVD decomposition for face recognition across pose and illumination. Another advantage of the proposed method approach over Tensorfaces is that, the proposed approach provides explicit sparse representations for each subject and each visual domain, which can be used for subject classification and domain estimation. Instead, Tensorfaces performs subject classification through exhaustive projections and matchings. Another work similar to Tensorfaces is discussed in [11], where a bilinear analysis is presented for face matching across domains. In [11], a 2-mode SVD decomposition is first performed and then a Gaussian mixture model is employed to classify subjects. Tensorfaces can be considered as an extension of this work to enable multilinear analysis to face images. The remainder of the paper is organized as follows: Section II discusses some details about sparse decomposition and multilinear image analysis. In Section III, we formulate the domain adaptive dictionary learning problem for face recognition. In Section IV, we present the proposed domain adaptive dictionary learning approach, which consists of algorithms to learn a domain base dictionary, and perform domain invariant sparse coding. Experimental evaluations are given in Section V on two public face datasets. Finally, Section VI concludes the paper.

A. Sparse Decomposition
Sparse signal representations have recently drawn much attention in vision, signal and image processing [12], [13], [14], [15], [16]. This is mainly due to the fact that signals and images of interest can be sparse in some dictionary. Given an over-complete dictionary D and a signal y, finding a sparse representation of y in D entails solving the following optimization problem where the 0 sparsity measure x 0 counts the number of nonzero elements in the vector x. Problem (1) is NP-hard and cannot be solved in a polynomial time. Hence, approximate solutions are usually sought [14], [17], [18], [19]. The dictionary D can be either based on a mathematical model of the data [14] or it can be trained directly from the data [20]. It has been observed that learning a dictionary directly from training rather than using a predetermined dictionary (such as wavelet or Gabor) usually leads to better representation and hence can provide improved results in many practical applications such as restoration and classification [12], [13].
Various algorithms have been developed for the task of training a dictionary from examples. One of the most commonly used algorithms is the K-SVD algorithm [21]. Let Y be a set of N input signals in a n-dimensional feature space Y = [y 1 ...y N ], y i ∈ R n . In K-SVD, a dictionary with a fixed number of K items is learned by finding a solution iteratively to the following problem: are the sparse codes of input signals Y, and T specifies the sparsity that each signal has fewer than t items in its decomposition. Each dictionary item d i is l 2 -normalized.

B. Multilinear Image Analysis
Linear methods are popular in facial image analysis, such as principal components analysis (PCA) [22], independent component analysis (ICA) [23], and linear discriminant analysis (LDA) [24]. These conventional linear analysis methods work best when variations in domains, such as pose and illumination, are not present. When any visual domain is allowed to vary, the linear subspace representation above does not capture such variation well.
Under the assumption of Lambertian reflectance, Basri and Jacobs [25] showed that images of an object obtained under a wide variety of lighting conditions can be approximated accurately with a 9-dimensional linear subspace. [26] utilizes the fact that 2D harmonic basis images at different poses are related by close-form linear transformations [27], [28], and extends the 9-dimensional illumination linear space with additional pose information encoded in a linear transformation matrix. The success of these methods suggests the feasibility of decomposing a face image into separate representations for subject and individual domains, e.g. associated pose and illumination, through multilinear algebra.
A multilinear image analysis approach, called Tensorfaces, has been discussed in [7], [9], [10]. Tensor is a multidimensional generalization of a matrix. An N -order tensor D is an N -dimensional matrix comprising N spaces. N -mode SVD, illustrated in Fig. 2, is an extension of SVD that decomposes the tensor as the product of N -orthogonal spaces, where Tensor Z, the core tensor, is analogous to the diagonal singular value matrix in SVD. Mode matrix U n contains the orthonormal vectors spanning the column space of mode-n flattening of D, i.e., the rearranged tensor elements that form a regular matrix [7]. Consider the illustrative example presented in [7]. Given faces images of 28 subjects, in 5 poses, 3 illuminations and 3 expressions, and each image contains 7943 pixels, we obtain a face tensor D of size 28 × 5 × 3 × 3 × 7943. Suppose we apply a multilinear analysis to the face tensor D using the 5-mode decomposition as (3).
where the 28 × 5 × 3 × 3 × 7943 core tensor Z governs the interaction between the factors represented in the 5 mode matrices, and each of the mode matrix U n represents subjects and respective domains. For example, the k th row of the 28 × 28 mode matrix U subject contains the coefficients for subject k, and the j th row of 5×5 mode matrix U pose contains the coefficients for pose j.
Tensorfaces perform subject classification through exhaustive projections and matchings. In the above examples, from the training data, each subject is represented with a 28-sized vector of coefficients to the 28 × 5 × 3 × 3 × 7943 base tensor in (4) One can then obtain the basis tensor for a particular pose j, illumination l, and expression e as a 28 × 1 × 1 × 1 × 7943 sized subtensor B j,l,e . The subject coefficients of a given unknown face image are obtained by exhaustively projecting this image into a set of candidate basis tensors for every j, l, e combinations. The resulting vector that yields the smallest distance to one of the rows in U pose is adopted as the coefficients for the subject in the test image. In a similar way, one can obtain the coefficient vectors for pose and illumination associated with such a test image.

III. PROBLEM FORMULATION
In this section, we formulate the domain adaptive dictionary learning (DADL) approach for face recognition. It is noted that our approach is general and applicable to both image and nonimage data. Let Y denote a set of N signals (face images) in an n-dim feature space Y = [y 1 , ..., y N ], y i ∈ R n . Given that face images are from K different subjects [S 1 , · · · , S K ], in J different poses [P 1 , · · · , P J ], and under L different illumination conditions [I 1 , · · · , I L ], Y can be arranged in six different forms as shown in Fig. 4. We assume here that one image is available for each subject under each pose and illumination, i.e., N = K × J × L.   Fig. 4. Each dictionary atom is in the R n space.
Any of the six forms in Fig. 4 can be transformed into another through a sequence of vector transpose operations. As illustrated in Fig. 3, a vector transpose operation is to consider (stacked) image vectors in Fig. 4 as values and perform typical matrix transpose operation. For simplicity, we define six aggregated vector transpose operations {T i } 6 i=1 . For example, T i transforms an input matrix, which is in any of the six forms, into the i-th form defined in Fig. 4.
Let y jl k be a face image of subject S k in pose P j under illumination I l . The dictionary adapted to pose P j and illumination I l is expressed as  y jl k can be sparsely represented using this dictionary as, where the subject sparse codes b k are independent of both P j and I l . In this way, we can represent Fig. 4 in a compact matrix form as shown in (5).
The proposed domain adaptive dictionary model is built as follows, • We learn a base dictionary D that is independent of subjects and domains. • We learn a sparse representation over the base dictionary for each visual domain, e.g., a specific pose or illumination condition. • We express the dictionary adapted to a specific domain as sparse linear combinations of the base dictionary using sparse representation of the domain under consideration.. • We learn for each subject a domain invariant sparse representation. We now provide the details of solutions to the following two problems • How to learn a base dictionary that is independent of subject and domains. • Given an input face image and the base dictionary, how to obtain the sparse representation for the associated pose and illumination, and the domain invariant sparse representation for the subject.

IV. DOMAIN ADAPTIVE DICTIONARY LEARNING
In this section, we first show, given a domain base dictionary D, sparse coefficient matrices are equivalent across different equations in (5). Then, we present algorithms to learn a domain base dictionary D, and perform domain invariant sparse coding.

A. Equivalence of Six Forms
To learn a domain base dictionary D, we first need to establish the following proposition.
Proposition: Given a domain base dictionary D, matrices in all six equations in (5) are equivalent, and so are matrices Fig. 4 are different only in the row order. We assume a permutation matrix P 16 will permutate the rows of Y 1 into Y 6 , i.e., P 16 Y 1 = Y 6 . Through a dictionary learning process, e.g., k-SVD [21], we obtain a dictionary D 1 and the associated sparse code matrix B 1 for Y 1 . Y 1 can be reconstructed as Y 1 = D 1 B 1 . We change the row order of D 1 according to P 16 without modifying the actual atom value as D 6 = P 16 D 1 . We decompose Y 6 using D 6 as Y 6 = D 6 B 6 , i.e., P 16 Y 1 = P 16 D 1 B 6 , and we have B 1 = B 6 .
Then we show that matrices A i , B i and C i in (5a) and (5b) are equivalent. If we stack all the images from the same subject under the same pose but different illumination as a single observation, we can consider Y 2 = Y T 1 . By assuming a bilinear model, we can represent 2. apply B to (5a) and solve via k-SVD 3. apply A to (5d) and solve via k-SVD min 4. apply C to (5e) and solve via k-SVD min Algorithm 1: Domain base dictionary learning.
and B i are equivalent in (5a) and (5b). As both equations share a bilinear map D T3 C i , with a common base dictionary D, matrices C i are also equivalent in (5a) and (5b). Finally, we show matrices A i and C i in (5a) and (5f) are equivalent. We have shown in (5a) and (5f) that matrices B i are equivalent.
are different only in the row order. We can use the bilinear model argument made above to easily show that matrices A i and C i are equivalent in (5a) and (5f).
Through the transitivity of equivalence, we can further show matrices A i in all six equations in (5) are equivalent, and so are matrices B i and C i . We drop the subscripts in subsequent discussions and denote them as A, B and C.

B. Domain Invariant Sparse Coding
As matrices A, B and C are equivalent across all six forms in (5), we propose to learn the base dictionary D using Algorithm 1 given below. Algorithm 1 is designed as an iterative method, and each iteration consists of several typical sparse dictionary learning problems. Thus, this algorithm is flexible and can rely on any sparse dictionary learning methods. We adopt the highly efficient dictionary learning method, k-SVD [21]. It is noted that we can easily omit one domain aspect through dictionary "marginalization". For example, after learning the based dictionary D, we can marginalize over illumination sparse codes matrix C and adopt [D T3 C] T2 as the base dictionary for pose domains only.
With the learned base dictionary D, we can perform domain invariant sparse coding as shown in Algorithm 2. This Input: an input image y, domain base dictionary D, sparsity level T a , T b , T c Output: sparse representation vector for pose a, illumination c, subject b begin Initialization stage: 1. Initialize domain sparse code vector a and c with random values; Sparse coding stage: repeat 2. apply a and c to (5a) and obtain b via any pursuit algorithm, . apply b and c to (5d) and obtain a via any pursuit algorithm, min . apply a and b to (5e) and obtain c via any pursuit algorithm, min sparse codes for the input pose: a, sparse codes for the input illumination: c; end Algorithm 2: Domain invariant sparse coding for a face image.
algorithm accepts any pursuit algorithms, such as OMP [18], [19]. Through this algorithm, an input face image can be decomposed into sparse representations for the associated pose and illumination, and a domain invariant sparse representation for the subject.
Convergence of Algorithms 1 and 2 can be established using the convergence results of k-SVD discussed in [21]. The convergence of both algorithms depends on the success of pursuit algorithms involved in each iteration step. We have observed empirical convergence for both Algorithm 1 and 2 in all the experiments reported below.

V. EXPERIMENTAL EVALUATION
This section presents experimental evaluations on two public face datasets: the CMU PIE dataset [29] and the Extended YaleB dataset [30].

B. Convergence of Domain Invariant Sparse Coding
We demonstrate here the convergence of the proposed domain invariant sparse coding in Algorithm 2 over a base dictionary learned using Algorithm 1. We first learn the domain base dictionary D 10 using Algorithm 1, and also obtain the associated domain matrices A, B and C. The matrix A consists of 10 columns and each column is a unique sparse representation for one of the 10 poses. The matrix B consists of 68 columns, and each column describes one of the 68 subjects. The matrix C consists of 21 columns, and each column describes one of the 21 illumination conditions. We observe no significant reconstruction improvements from the learned base dictionary after 2 iterations of Algorithm 1.
Given a face image (s43, c29, f 05), i.e., subject s43 in pose c29 under illumination f 05, Fig. 6a, 6b and 6c show the decomposed sparse representations for subject s43, pose c29 and illumination f 05 after 1, 2 and 100 iterations of Algorithm 2 respectively. We can notice that the decomposed sparse codes (color red) converge to the learned model codes (color blue) in A, B and C. As shown in Fig. 8a, we observe convergence after 4 iterations.
[31] proposed a Tensor k-SVD method, which is similar to Tensorfaces but replaces the N -mode SVD with k-SVD to perform multilinear sparse decomposition. Using the Tensor k-SVD method, we are able to learn a Tensor k-SVD dictionary and the associated domain matrices. As the Tensor k-SVD method is designed for data compression, it is not discussed in [31] how to decompose a single image into separate spare coefficient vectors over such learned Tensor k-SVD dictionary. We adopt a learned Tensor k-SVD dictionary as the base dictionary for domain invariant sparse coding using Algorithm 2. As shown in Fig.6d, the decomposed sparse codes do not converge well to the learned model codes. It indicates  that Algorithm 2 performs an inconsistent decomposition over the Tensor k-SVD dictionary. Therefore, a base dictionary learned from Algorithm 1 is required by the proposed domain invariant sparse coding in Algorithm 2 to enforce a consistent multilinear sparse decomposition.
We further decompose face images (s43, c27, f 13) and (s01, c27, f 13) over D 10 . As shown in Fig. 7, even pose c27 is unknown to D 10 , the decomposed sparse codes for subjects s43 and s01, and illumination f 13 still converge to the learned models. By comparing the pose codes in Fig. 7a and 7b, we notice that the unknown pose c27 is represented as a sparse linear combination of known poses in a consistent way. We observe convergence after about 10 iterations for both cases, as shown in Fig.8b and Fig.8c.

C. Domain Composition
Using the proposed trilinear sparse decomposition over a base dictionary as illustrated in Algorithm 2, we extract from a face image the respective sparse representations for subject, pose and illumination. We can translate a subject to a different pose and illumination by composing the corresponding subject and domain sparse codes over the base dictionary. As discussed in Sec. II-B, Tensorfaces also enables the decomposition of a face image into separate coefficients for the subject, pose and illumination through exhaustive projections and matchings. We adopt the Tensorfaces method here for a fair comparison in our domain composition experiments.    poses are present in the training data, we extract the subject (sparse) codes for s01 from each of them. Then we extract the pose codes for c27 (frontal) and the illumination codes for f 05 from an image of subject s43. It is noted that, for such known subject cases, the composition (s01, c27, f 05) through both DADL and Tensorfaces provides good reconstructions to the ground truth image. The reconstruction using DADL is clearer than the one using Tensorfaces.
In Fig. 9b, we first extract the subject codes for s43, which is an unknown subject to D 34 . Then we extract the pose codes and the illumination codes from the set of images of s01 in Fig. 9a. In this unknown subject case, the composition using our DADL method provides significantly more accurate reconstruction to the groudtruth images than the Tensorfaces method. The central assumption in the literature on sparse representation for faces is that the test face image should be represented in terms of training images of the same subject [32], [33]. As s43 is unknown to D 34 , therefore, it is expected that the reconstruction of the subject information is through a linear combination of other known subjects, which is an approximation but not exact.
In Fig. 9c, the base dictionary D 10 is used in the DADL experiments, and the same training data and sparsity values for D 10 are used in the corresponding Tensorfaces experiments. We first extract the subject codes for s43. Then we extract the pose codes for pose c22, c05 and c27, which are unknown poses to the training data. Through domain composition, for such unknown pose cases, we obtain more acceptable reconstruction to the actual images using DADL than Tensorfaces. This indicates that, using the proposed DADL method, an unknown pose can be much better approximated in terms of a set of observed poses.
2) Illumination Normalization: In Fig. 10a, we use frontal faces from subject s28, which is known to D 34 , under different illumination conditions. For each image, we first isolate the codes for subject, pose and illumination, and then replace the illumination codes with the one for f 11. If f 11 is observed in the training data, the illumination codes for f 11 can be obtained during training. Otherwise, the illumination codes for f11 can be extracted from any face image under f 11 illumination. It is shown in Fig. 10a that, for such known subject cases, after removing the illumination variation, we can obtain a reconstructed image close to the ground truth image using both DADL and Tensorfaces.  In each corresponding Tensorfaces experiment, we adopt the same training data and sparsity values used for the DADL base dictionary for a fair comparison. When a subject or a pose is unknown to the training data, the proposed DADL method provides significantly more accurate reconstruction to the ground truth images.
Subject s43 in Fig. 10b is unknown to D 34 . The composed images from DADL exhibit significantly more accurate subject, pose and illumination reconstruction than Tensorfaces. As discussed before, the reconstruction to the subject here is only an approximation but not exact.

D. Pose and Illumination Invariant Face Recognition
1) Classifying PIE 68 Faces using D 4 and D 10 : Fig. 11 shows the face recognition performance under combined pose and illumination variation for the CMU PIE dataset. To enable the comparison with [8], we adopt the same challenging setup as described in [8]. In this experiment, we classify 68 subjects in three poses, frontal (c27), side (c05), and profile (c22), under all 21 lighting conditions. We select one of the 3 poses as the gallery pose, and one of the remaining 2 poses as the probe pose, for a total of 6 gallery-probe pose pairs. For each pose pair, the gallery is under the lighting condition f 11 as specified in [8], and the probe is under the illumination indicated in the table. Methods compared here include Tensorface [7], [9], SMD [8], and our method DADL. DADL-4 uses the dictionary D 4 and DADL-10 uses D 10 . In both DADL-4 and DADL-10 setups, three testing poses c27, c05, and c22 are unknown to the training data. It is noted that, to the best of our knowledge, SMD reports the best recognition performance in such experimental setup. As shown in Fig. 11, among 4 out of 6 Gallery-Probe pose pairs, the proposed DADL-10 is better or comparable to SMD.  In each corresponding Tensorfaces experiment, we adopt the same training data and sparsity values used for the DADL base dictionary for a fair comparison. When a subject is unknown to the training data, the proposed DADL method provides significantly more accurate reconstruction to the ground truth images.
based on the stereo matching distance between each pair of gallery-probe images. The stereo matching distance becomes more robust when the pose variation between such image pair decreases. However, the proposed DADL classifies faces based on subject codes extracted from each image alone. The robustness of the extracted subject codes only depends on the capability of the base dictionary to reconstruct such a face. This explains why our DADL method significantly outperforms SMD for more challenging pose pairs, e.g., Profile-Frontal pair with 62 o pose variation; but performs worse than SMD for easier pairs, e.g., Frontal-Side with 16 o pose variation. It can be observed in Fig. 9c that an unknown pose can be approximated in terms of a set of observed poses. By representing three testing poses through four training poses in D 4 , instead of ten poses in D 10 , we obtain reasonable performance degradations but with 60% less training data.
Though the Tensorface method shares a similar multilinear framework to DADL, as seen from Fig. 11, it only handles limited pose and illumination variations.
2) Classifying Extended YaleB using D 32 : We adopt a similar protocol as described in [34]. In the Extended YaleB dataset, each of the 38 subjects is imaged under 64 lighting conditions. We split the dataset into two halves by randomly selecting 32 lighting conditions as training, and the other half for testing. Fig. 12 shows the illumination variation in the testing data. When we learn D 32 using Algorithm 1, we also obtain the sparse codes for each subject. During testing, we extract the subject codes from each testing face image and classify it based on the best match in subject codes learned from the training data. As shown in Table I, the proposed DADL method outperforms other state-of-the-art sparse representation methods (The results for other compared methods are taken from [34]). When the extreme illumination conditions are included, we obtain an average recognition rate 98.67%. By excluding two extreme illumination condition f 34 and f 35, we obtain an average recognition rate 99.7%.

E. Mean Code and Error Analysis
As discussed in Sec. II-B, the Tensorface method shares a similar multilinear framework to the proposed DADL method. However, we showed through the above experiments that the proposed method based on sparse decomposition significantly outperforms the N -mode SVD decomposition for face recognition across pose and illumination. In this section, we analyze in more detail the behaviors of the proposed DADL and Tensorfaces, by comparing subject and domain codes extracted from a face image using these two methods.
For the experiments in this section, we adopt the base dictionary D 10 for DADL, and the same training data and sparsity values of D 10 for Tensorfaces to learn the core tensor and the associated mode matrices. The same testing data is used for both methods, i.e., 68 subjects in the PIE dataset under 21 illumination conditions in the c27 (frontal), c05 (side) and c22 (profile) poses, which are three unseen poses not present in the training data. Fig. 13 and Fig. 14 shows the mean subject codes of subject s1 and s2 over 21 illumination conditions in each of the three testing poses, and the associated standard errors. In each of the two figures, we compare the first row, the subject codes from DADL, with the second row, the subject codes from Tensorfaces. We can easily notice the following: first, the subject      f2 f5 f6 f12 f14 f15 f16 f17 f18 f19 f21 f26 f29 f31 f34 f35 f37 f38 f40 f41 f44' f45 f46 f47 f50 f52 f53 f55 f58 f59 f62  codes extracted using DADL are more sparse; second, DADL subject codes are more consistent across pose; third, DADL subject codes are more consistent across illumination, which is indicated by the smaller standard errors. By comparing Fig. 13 with Fig. 14, we also observe that the DADL subject codes are more discriminative. Table II further shows the square root of the pooled variances of subject codes for all 68 subjects over 21 illumination conditions in each of the three testing poses. The significantly smaller variance values obtained using DADL indicate the more consistent sparse representation of subjects decomposed from face images. Therefore, face recognition using DADL subject codes significantly outperforms recognition using Tensorfaces subject codes.

F. Pose and Illumination Estimation
In Section V-D, we report the results of experiments over subject codes using base dictionaries D 10 and D 4 . While generating subject codes, we simultaneously obtain pose codes and illumination codes. Such pose and illumination codes can be used for pose and illumination estimation. In Fig. 15, we show the pose and illumination estimation performance on the PIE dataset using the pose and illumination sparse codes through both DADL and Tensorfaces. The proposed DADL method exhibits significantly better domain estimation

VI. CONCLUSION
We presented an approach to learn domain adaptive dictionaries for face recognition across pose and illumination domain shifts. With a learned domain base dictionary, an unknown face image is decomposed into subject codes, pose codes and illumination codes. Subject codes are consistent across domains, and enable pose and illumination insensitive face recognition. Pose and illumination codes can be used to estimate the pose and lighting condition of the face. The proposed method can be generalized for multilinear face image analysis, however, more experimental validations are needed. We also plan to evaluate the usefulness of our domain adaptive dictionary learning framework in applications other than face recognition.