codes,and then learns the orthogonal matrix based on the resulting binary codes:thirdly.IsoHash has an explicit objective function to optimize,but ITQ uses a two-step heuristic strategy whose goal cannot be formulated by a single objective function;fourthly,to learn the orthogonal matrix, IsoHash uses Lift and Projection or Gradient Flow,but ITQ uses Procruster method which is much slower than IsoHash.From the experimental results which will be presented in the next section,we can find that IsoHash can achieve accuracy comparable to ITQ with much faster training speed. 4 Experiment 4.1 Data Sets We evaluate our methods on two widely used data sets,CIFAR [16]and LabelMe [28]. The first data set is ClFAR-10 [16]which consists of 60,000 images.These images are manually labeled into 10 classes,which are airplane,automobile,bird,cat,deer,dog,frog,horse,ship,and truck.The size of each image is 32x32 pixels.We represent them with 256-dimensional gray-scale GIST descriptors [24] The second data set is 22K LabelMe used in [23.28]which contains 22.019 images sampled from the large LabelMe data set.As in [28].The images are scaled to 32x32 pixels,and then represented by 512-dimensional GIST descriptors [241. 4.2 Evaluation Protocols and Baselines As the protocols widely used in recent papers [7,23,25,31],Euclidean neighbors in the original s- pace are considered as ground truth.More specifically,a threshold of the average distance to the 50th nearest neighbor is used to define whether a point is a true positive or not.Based on the Euclidean ground truth,we compute the precision-recall curve and mean average precision(mAP)[7,21].For all experiments,we randomly select 1000 points as queries,and leave the rest as training set to learn the hash functions.All the experimental results are averaged over 10 random training/test partitions. Although a lot of hashing methods have been proposed,some of them are either supervised [23] or semi-supervised [29].Our IsoHash method is essentially an unsupervised one.Hence,for fair comparison,we select the most representative unsupervised methods for evaluation,which contain PCAH [7],ITQ [7],SH [31],LSH [1],and SIKH [25].Among these methods,PCAH,ITQ and SH are data-dependent methods,while SIKH and LSH are data-independent methods. All experiments are conducted on our workstation with Intel(R)Xeon(R)CPU X7560@2.27GHz and 64G memory 4.3 Accuracy Table 1 shows the Hamming ranking performance measured by mAP on LabelMe and CIFAR.It is clear that our IsoHash methods,including both IsoHash-GF and IsoHash-LP,achieve far better performance than PCAH.The main difference between IsoHash and PCAH is that the PCAH di- mensions have anisotropic variances while IsoHash dimensions have isotropic variances.Hence, the intuitive viewpoint that using the same number of bits for different projected dimensions with anisotropic variances is unreasonable has been successfully verified by our experiments.Further- more,the performance of IsoHash is also comparable,if not superior,to the state-of-the-art methods, such as ITQ. Figure 1 illustrates the precision-recall curves on LabelMe data set with different code sizes.The relative performance in the precision-recall curves on CIFAR is similar to that on LabelMe.We omit the results on CIFAR due to space limitation.Once again,we can find that our IsoHash methods can achieve performance which is far better than PCAH and comparable to the state-of-the-art. 4.4 Computational Cost Table 2 shows the training time on CIFAR.We can see that our IsoHash methods are much faster than ITQ.The time complexity of ITQ also contains two parts:the first part is PCA which is the samecodes, and then learns the orthogonal matrix based on the resulting binary codes; thirdly, IsoHash has an explicit objective function to optimize, but ITQ uses a two-step heuristic strategy whose goal cannot be formulated by a single objective function; fourthly, to learn the orthogonal matrix, IsoHash uses Lift and Projection or Gradient Flow, but ITQ uses Procruster method which is much slower than IsoHash. From the experimental results which will be presented in the next section, we can find that IsoHash can achieve accuracy comparable to ITQ with much faster training speed. 4 Experiment 4.1 Data Sets We evaluate our methods on two widely used data sets, CIFAR [16] and LabelMe [28]. The first data set is CIFAR-10 [16] which consists of 60,000 images. These images are manually labeled into 10 classes, which are airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. The size of each image is 32×32 pixels. We represent them with 256-dimensional gray-scale GIST descriptors [24]. The second data set is 22K LabelMe used in [23, 28] which contains 22,019 images sampled from the large LabelMe data set. As in [28], The images are scaled to 32×32 pixels, and then represented by 512-dimensional GIST descriptors [24]. 4.2 Evaluation Protocols and Baselines As the protocols widely used in recent papers [7, 23, 25, 31], Euclidean neighbors in the original s￾pace are considered as ground truth. More specifically, a threshold of the average distance to the 50th nearest neighbor is used to define whether a point is a true positive or not. Based on the Euclidean ground truth, we compute the precision-recall curve and mean average precision (mAP) [7, 21]. For all experiments, we randomly select 1000 points as queries, and leave the rest as training set to learn the hash functions. All the experimental results are averaged over 10 random training/test partitions. Although a lot of hashing methods have been proposed, some of them are either supervised [23] or semi-supervised [29]. Our IsoHash method is essentially an unsupervised one. Hence, for fair comparison, we select the most representative unsupervised methods for evaluation, which contain PCAH [7], ITQ [7], SH [31], LSH [1], and SIKH [25]. Among these methods, PCAH, ITQ and SH are data-dependent methods, while SIKH and LSH are data-independent methods. All experiments are conducted on our workstation with Intel(R) Xeon(R) CPU X7560@2.27GHz and 64G memory. 4.3 Accuracy Table 1 shows the Hamming ranking performance measured by mAP on LabelMe and CIFAR. It is clear that our IsoHash methods, including both IsoHash-GF and IsoHash-LP, achieve far better performance than PCAH. The main difference between IsoHash and PCAH is that the PCAH di￾mensions have anisotropic variances while IsoHash dimensions have isotropic variances. Hence, the intuitive viewpoint that using the same number of bits for different projected dimensions with anisotropic variances is unreasonable has been successfully verified by our experiments. Further￾more, the performance of IsoHash is also comparable, if not superior, to the state-of-the-art methods, such as ITQ. Figure 1 illustrates the precision-recall curves on LabelMe data set with different code sizes. The relative performance in the precision-recall curves on CIFAR is similar to that on LabelMe. We omit the results on CIFAR due to space limitation. Once again, we can find that our IsoHash methods can achieve performance which is far better than PCAH and comparable to the state-of-the-art. 4.4 Computational Cost Table 2 shows the training time on CIFAR. We can see that our IsoHash methods are much faster than ITQ. The time complexity of ITQ also contains two parts: the first part is PCA which is the same 7