## Introduction

Distance/similarity measure between images plays a fundamental role in many computer vision applications, e.g., person verification

[Hirzer et al.2012], matching heterogeneous data [Zhai et al.2012], and multimedia retrieval [Chechik et al.2012]. Conventional Mahalanobis distance metric learning models, which aim to seek a linear transformation by pulling the distances between similar pairs while pushing the distances between dissimilar pairs, are theoretically appealing in producing discriminative similarity or distance measure from the given training data

[Weinberger, Blitzer, and Saul2005a]. However, these approaches usually are performed in the original data space or the hand-engineered feature space (i.e. representation), and thus are limited in capturing variety of image appearance and handling complicated nonlinear manifold.In this paper, we investigate the possible interaction between feature learning and distance metric learning, and address the very challenging task of person verification (i.e. matching individuals across cameras). Figure 1 shows some examples of this task, where the pedestrians across cameras distinctly vary in appearance and structure caused by pose, lighting and view angle changes. The proposed framework is built based on the convolutional neural network (CNN) [LeCun et al.1989], which jointly optimizes the human representation as well as the proper distance metric to robustly match individuals against various real challenges. We call this framework DARI (i.e. Distance metric And Representation Integration).

We aim at preserving similarity of the same person while discriminating the different individuals, and thus define DARI in the form of maximizing relative distance. Specifically, we organize the training images into triplet units, and each unit contains one person image and the matched/mismatch references. For all triplet units, the objective of DARI is to maximize the distance disparity between the matched pairs and the mismatched pairs. In particular, DARI seeks optimal solutions for both feature representation and distance metric, while the existing approaches of person verification [Xu et al.2013, Liu et al.2012]

focuses on only either of the two components. We adopt the deep CNN architecture to extract the discriminative features from the input images, where the convolutional layers, max-pooling operators, and one full connection layer are stacked up. And the Mahalanobis distance matrix is applied with the generated feature as the matching metric. Due to the positive semi-definite requirement for the Mahalanobis metric, directly optimizing the metric matrix is computational intensive. We propose to decompose the Mahalanobis matrix inspired by

[Mignon and Jurie2012], and further factorize the matrix into a fully-connected layer on the top of our deep architecture. In this way, the distance metric is seamlessly integrated with the image feature represented by the other layers of neural networks. The joint optimization can be then efficiently achieved via the standard backward propagation. Therefore, by means of the nonlinearity learning of deep neural networks, DARI is capable of representing the complicated transformation to identify the people in the wild.To scale up our approach to the large amount of training data, we implement the training in a batch-process fashion. In each round of training, we randomly select a relatively small number (say

) of images, and use them to organize the triplet units. By taking the triplets as the inputs, we update the model parameters by the stochastic gradient descent (SGD) algorithm

[LeCun et al.1998]. Another arising issue is that the triplet organization cubically enlarges the number (say ) of training samples, as one image can be included into more than one triplet. To overcome it, we calculate the gradients on the images instead of the produced triplets, and thus reduce the computation cost by making it only depends on the number of the selected images.The key contribution of this paper is a novel end-to-end framework that naturally fuses the concept of feature learning and metric learning via the deep neural networks. To the best of our knowledge, such an approach is original to the community. On several challenging benchmarks for person verification (e.g., CUHK03 [Li et al.2014],CUHK01[Li, Zhao, and Wang2012] and iLIDS [Zheng, Gong, and Xiang2013]), our DARI framework demonstrates superior performances over other state-of-the-art approaches.

## Related Work

A number of approaches, e.g., local metric learning and kernelized metric learning, have been suggested to learn multiple or nonlinear metrics from training data with complicated nonlinear manifold structure. In local metric learning, local metrics can be learned independently for each region or by considering the data manifold structure [Noh, Zhang, and Lee2010, Wang, Kalousis, and Woznica2012, Weinberger, Blitzer, and Saul2005b]. In kernelized metric learning, a kernel function is exploited to implicitly embed instances into the reproducing kernel Hilbert space (RKHS), and a Mahalanobis distance metric is then learned in the RKHS space [Wang et al.2011]. Actually, kernelized method with Gaussian RBF kernel can also be treated as local learning approach. As pointed out in [Bengio2009], local learning are also shallow models, and generally are insufficient in coping with highly varying appearance and deformations of images. Another efficient local distance metric learning [Yang et al.2006] was also proposed for classification and retrieval. To handle heterogeneous data, [Xiong et al.2012]

propose a method using a random forest-based classifier to strengthen the distance function with implicit pairwise position dependence.

On the other hand, deep convolutional models have been intensively studied and achieved extremely well performance. Compared with the multiple layer perceptron, CNN contains much less parameters to be learned, and can be efficiently trained using stochastic gradient descent. With the increasing of large scale training data and computational resources, deeper CNN and novel regularization methods had been developed, and deep CNN has gained great success in many visual recognition tasks, e.g., image classification

[Krizhevsky, Sutskever, and Hinton2012a], object detection [Szegedy, Toshev, and Erhan2013], and scene labeling [Pinheiro and Collobert2014].Despite the success of deep learning in variety of vision tasks, little studies were conducted on metric learning with deep architecture. Chopra et al.

[Chopra, Hadsell, and LeCun2005]suggested a energy-based model (EBM) for discriminative similarity metric learning for image pairs. Stacked restricted Boltzmann machines (RBMs) had also been exploited to learn nonlinear transformation for data visualization and supervised embedding

[Min et al.2010]. Cai et al. [Cai et al.2010]proposed a deep nonlinear metric learning method by combining logistic regression and independent subspace analysis. Hu et al.

[Hu, Lu, and Tan2014] adopted the forward multi-layer neural network to learn deep metric for hand-crafted features. Compared with these approaches, the proposed DARI model considers the prominence of CNN in capturing salient and incorporates the Mahalanobis distance with the generated image features into one optimization target for distance metric and representation integration.One approach close to ours was proposed by Wang et al. [Wang et al.2014]

, which addresses the triplet-based similarity learning for image retrieval. However, our work have significant differences with that work. First, we derive our formulation from a novel angle, i.e. integrating feature learning and distance metric learning. Second, our learning method has advantage in the triplet generation and the batch-based gradient descent learning. Specifically, given

training triplets containing distinct images (), their algorithm optimizes with forward and backward propagations, while only rounds is required for our approach because we derive to calculate the gradient over the images. Last, our deep architecture is specifically designed (only two conv layers are used) and we train our model from scratch, while they utilized the Alex’s model [Krizhevsky, Sutskever, and Hinton2012b]that is pre-trained on the ImageNet.

## Framework

### Primal Formulation

Given a fixed feature representation, metric learning is to learn a distance function by satisfying the constraint according to the label information of samples. Here we define our formulation via relative distance comparison based on the triplet-based constraint. As is discussed in [Ding et al.2015], the triplet models allows the images of one identity lying on a manifold while maximizing the margin between within-class pairs from between-class pairs, and tends to result in better tradeoff between adaptability and discriminability.

More precisely, the relative distance constraint is defined with a set of triplet units , in which is a pair of matched images (images of the same individual) and contains two mismatched images from the labeled image set with denoting the label. Let denote the metric matrix and denote the feature representations of the th image learned by the feature network with the network parameters . Then Mahalanobis distance between and using the CNN features can be written as follows:

(1) |

where denotes the feature difference between the image and . For each training triplet in , the desired distance should satisfy: . Let denote , we turn this relative constraints into the minimization of the following hinge-loss like objective function where acts as a regularization term as in [Shen et al.2012].

In the following, we use to denote for notation simplicity. By the definition of , we get the following objective functions:

(2) |

An intuitive solution to Eqn. (2) is to iteratively optimize and with either of them fixed. It is, however, computationally expensive, as the PSD projection is necessarily imposed once is updated to ensure to be positive semi-definite. In this work, to overcome this issue we propose an unified solution by incorporating into the deep neural networks.

### Deep Architecture

We factorize the metric matrix into as satisfies the positive semi-definite constraint. The distance measure in Eqn. (1) can be then transformed as,

(3) |

The objective function in Eqn. (2) can be then reformulated with , as

(4) | ||||

Thus, we can take as a linear transform on the output of the CNN-based feature representation. In literature, [Weinberger, Blitzer, and Saul2005a] [Mignon and Jurie2012] also proposed to decompose the Mahalanobis distance matrix for simplifying the distance metric learning. They attempt to pursue a low-dimensional projection in Euclidean space which embeds the distance metric while ensuring the PSD constraint. However, their solution are complicated requiring additional hypothesis. In this work, we implement a fully connected layer to represent , which is stacked over the layers representing image features, making the distance metric tightly combined with the deep neural network. Specifically, we treat

as the neuron weight of the layer, and the network can represent

by taking as the input. Then, the feature learning and the distance metric can be thus jointly optimized in an end-to-end way.In the following, we denote for notation simplicity. As the regularization term on will be automatically implemented by the built-in weight decay mechanisms in CNN networks, we neglect this part in the objective function.

(5) | ||||

Integrating the metric learning and feature learning into one CNN network yields several advantages. First, this leads to a good property of efficient matching. In particular, for each sample stored in a database, we can precomputed its feature representation and the corresponding decomposed Mahalanobis distance matrix. Then the similarity matching in the testing stage can be very fast. Second, it integrates feature learning and metric learning by building an end-to-end deep architecture of neural networks.

As discussed above, our model defined in Eqn. (5) jointly handles similarity function learning and feature learning. This integration is achieved by building a deep architecture of convolutional neural networks, which is illustrated in Figure 2

. Our deep architecture is composed of two sub-networks: feature learning sub-network and metric learning sub-network. The feature learning sub-network contains two convolution-RELU-pooling layers and one fully-connected layer. Both the pooling layers are max-pooling operations with the size of

and the stride size is set as

pixels. The first convolutional layer includes kernels of size with a stride of pixels. The second convolutional layer contains filters of size and the filter stride step is set aspixel. A fully-connected layer is followed and it outputs a vector of

dimensions. We further normalize the output vector of this fully-connected layer before it is fed to the metric learning sub-network by , where , denote the value of theth neuron before and after normalization respectively. Accordingly, the back propagation process accounts for the normalization operation using the chain rule during calculation of the partial derivatives. The metric learning sub-network includes only one fully-connected layer. The neural layer outputs

. In this way, the distance metric is tightly integrated with the feature representations, and they can be jointly optimized during the model training.## Learning Algorithm

Given a labeled dataset with classes (persons) and each class has images, then the number of all possible meaningful triplets is . Even for a dataset of moderate size, it is intractable to load all these triplets into the limited memory for the model training. To overcome this issue, we apply batch learning to optimize the parameters, in which the network parameters are updated by the gradient derived only from a small part of all the triplets in each iteration.

### Batch Process

In the batch learning process, we need to generate a subset of triplets from all the possible triplets in each iteration. The simplest method is to generate triplets randomly. However, this method makes the number of distinct images be approximately three times the number of the generated triplets because each triplet contains three images, and the likelihood of two triplets sharing the same image is very low. This triplet generation method is very inefficient because there are only a few distance constraints placed on the selected images in each iteration. Instead, to capitalize on the strength of relative distance comparison model, a more reasonable triplet generation method would be one that satisfies the two following conditions:

1. In each iteration, large number of triplets are generated from small number of images to ensure the selected images can be loaded to the memory while rich distance constraints are posed on these images;

2. When increased numbers of iterations are executed, all the possible relative distance constraints between any two classes should be considered in the training process.

These two principles lead to our proposed triplet generation scheme as follows. In each iteration, we select a fixed number of classes (persons), and construct the triplets only using these selected classes. More precisely, for each image in each class, we randomly construct a certain number of triplets with the matched reference coming from the same class and the mismatched references coming from the remaining selected classes. The complete mini-batch learning process is presented in Algorithm 1.

### Parameter Optimization

Under the mini-batch training framework, a key step is to calculate the gradient for the triplet set in each iteration. A straight method is to calculate the gradient for each triplet according to the loss function, and sum these gradients to get the overall gradient. But with this approach three separate memory units and a network propagation would be needed for every triplet. This is inefficient as there will be duplicated network propagation for the same image, recalling that for each batch we generate triplets from a known subset of images. We now show that there exists an optimized algorithm in which the computational load mainly depends on the number of distinct images rather than the number of the triplets.

It would be difficult to write the objective function in Eqn. (5) directly as the sum of image-based loss items because it takes the following form (for notation simplicity, we use to denote in the rest of the paper):

Fortunately, because the loss function for a specific triplet is defined by the outputs of the images in this triplet, the total loss can also be considered as follows, where represents the set of all the distinct images in the triplets and denote the size of the distinct images in the triplets.

By the derivative rule, we have the following equations, where represents the network parameters, represents the feature maps of the image at the layer and denote the partial derivative derived from image .

(6) |

(7) |

Eqn. 6 shows that the overall gradient is the sum of the image-based terms (image-based gradient). Eqn. 7 shows that the partial derivative with respect to the feature maps of each image can be calculated recursively. With Eqn. 6 and Eqn. 7, the gradients with respect to the network parameters can be obtained by summing the image based gradients using the network back propagation algorithm. The central premise is that we have computed the partial derivative of the output layer’s activation for every image, which can be easily obtained from Eqn. 5. Algorithm 2 gives the detailed process. This optimized algorithm has two obvious merits:

1. We can conveniently use exiting deep learning implementations such as Caffe

^{1}

^{1}1http://caffe.berkeleyvision.org/ to train our model.

2. The number of network propagation executions can be reduced to the number of distinct images in the triplets, a crucial advantage for large scale datasets.

## Evaluations

Datasets and Implementation details. We conduct our experiments using three challenging human verification datasets, i.e. CUHK03[Li et al.2014], CUHK01[Li, Zhao, and Wang2012] and iLIDS[Zheng, Gong, and Xiang2013] . All the images are resized to 250

100 for the experiment. The weights of the filters and the full connection parameters are initialized from two zero-mean Gaussian distributions with standard deviation 0.01 and 0.001 respectively. The bias terms were set with the constant 0. During the training, we select 60 persons to construct 4800 triplets in each iteration. Before feeding to the network, the images are mirrored with 0.5 probability and cropped to the size 230

80 at the center with a small random perturbation to augment the training data. We implement our learning algorithm based on the Caffe framework, where we revise the data layer and loss layer to generate the triplets and apply our loss function. We execute the code on a PC with GTX780 GPU and quad-core CPU. And stop the training process when there are less than 10 triplets whose distance constraints are violated, i.e. the distance between the matched pair is greater than the distance between the mismatched pair.Evaluation Protocol. We adopt the widely used cumulative match curve (CMC) approach [Gray, Brennan, and Tao2007] for quantitative evaluation. We follow the standard setting to randomly partition each dataset into training set and test set without overlap. In each testing, the test set is further divided into a gallery set and a probe set without overlap for 10 times. A rank rate is obtained for each time, and we use the average rate as the final result.

Component analysis. In order to demonstrate how the joint optimization of distance metric with feature representation contributes to performance, we implement a simplified model for comparison by discarding the distance metric learning(i.e. the last neural layer). In this implementation, we only optimize CNN-based feature representation by the back-propagation method.

Experiments on CUHK03 Dataset. This benchmark [Li et al.2014] is the largest one up to date, which contains 14096 images of 1467 pedestrians collected from 5 different pairs of camera views, making it an ideal place for deep learning. Each person is observed by two disjoint camera views and has an average of 4.8 images in each view. We follow the standard setting of using CUHK03 to randomly partition this dataset for 10 times without overlap, and a training set (including 1367 persons) and a test set (including 100 persons) are obtained. In each testing, the testing set is further randomly divided into a gallery set of 100 images (i.e. one image per person) and a probe set (including images of individuals from different camera views in contrast to the gallery set) without overlap for 10 times.

We introduce several types of state-of-the-arts in this experiment. First, we adopt Local Distance Metric Learning (LDM) [Guillaumin, Verbeek, and Schmid2009], the learning-to-rank method (RANK) [McFee and Lanckriet2010] for comparison, which learn distance metrics based on a fixed feature representation. Following their implementation, the handcrafted features of dense color histograms and dense SIFT uniformly sampled from patches are adopted. Two methods especially designed for person re-identification are introduced in this experiment: KISSME [Kostinger et al.2012], eSDC [Zhao, Ouyang, and Wang2013]. Moreover, we compare with a recently proposed deep learning method, DRSCH [Zhang et al.2015], FPNN [Li et al.2014] and IDLA [Ahmed, Jones, and Marks2015]. DRSCH [Zhang et al.2015] learns hashing code with regularized similarity for image retrieval and person re-identification. FPNN [Li et al.2014] learns pairs of filters to extract person representation and IDLA [Ahmed, Jones, and Marks2015] is also recently proposed deep learning method for person re-identification.

The results are shown in Fig. 3 (a). It is encouraging to observe that our approach achieves a new state-of-the-art on CUHK03. Note that without the joint optimization of distance metric and representation, the performance (i.e., “ Ours-nj”) degenerates from 55.4% to 45.4%.

Experiments on CUHK01 Dataset.

CUHK01 contains 971 individuals, each of which has two samples captured by two disjoint camera views. We partition this dataset into a training set and a testing set exactly following [Li et al.2014][Ahmed, Jones, and Marks2015]: 100 persons are used for testing and the remaining 871 persons for training. Each person has two images for each view and we randomly select one into the gallery set. Single-shot is adopted in the evaluation of CMC curve.

In addition to comparing with the methods adopted in the experiment on CUHK03, we introduce a recently proposed method which also addresses the interaction of representation learning and metric Learning (denoted as L + X) [Liao et al.2015].

Fig. 3 (b) shows the comparison of our DARI framework with other approaches. DARI achieves a new state of the art, with a rank-1 recognition rate of 65.95%. The gain of the joint optimization of distance metric and CNN-based feature is also clear on this dataset, 65.95% over 57.7%.

Cross-dataset Evaluation

The iLIDS dataset [Zheng, Gong, and Xiang2013] was constructed from video images captured in a busy airport arrival hall. It has 119 pedestrians, totaling 479 images. We conduct a challenging task, i.e. cross-dataset task using this dataset, which accords with the real-world surveillance applications. Specifically, we randomly split this dataset into a gallery set and a probe set: the gallery contains only one image of each pedestrian and the remaining images form the probe set. Our model is trained on CUHK03 and tested on this iLIDS dataset without fine-tuning the parameters.

We compare our DARI with several existing methods such as Xing’s [Xing et al.2002], and MCC [Globerson and Roweis2005]. They all use an ensemble of color histograms and texture histograms as the feature representation. Note that the results reported by these competing methods are generated by a different setting: both of the training and the testing data are from this dataset.

Fig. 3 (c) shows the quantitative results. Our superior performance over other approaches demonstrate the good generalization power of DARI. On the other hand, without incorporating Mahalanobis distance matrix, the performance (i.e. “Ours-nj” in ) clearly degenerates from 42.8% to 36.9%, which highlights the significance of the joint optimization of feature representation and distance metric. In the following, we further evaluate our approach under different implementation setting on iLIDS.

Data Augmentation Strategy. We crop the center of the images with random perturbation to augment the training data. This mechanism can effectively alleviate the over-fitting problems. Without this augmentation scheme, the top 1 performance drop by about 30 percent relatively.

Triplet Generation Scheme. We compared two generation strategy. In the first strategy, we select 60 persons for each iteration and only construct 60 triplets for these persons. In the second strategy, we select the same number of persons while constructing 4800 triplets for these persons. As expected by our analysis, the learning process of the first strategy is much slower than the second strategy and when the learning process of the second strategy converges in 7000 iterations, the performance of the first strategy only achieves about 70 percent of the second strategy.

## Conclusion

We have presented a novel deep learning framework incorporating Mahalanobis distance matrix with convolutional neural networks. In future work, we will extend our approach for larger scale heterogeneous data, thereby exploring new applications.

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