Supplemental Material for Can Visual Recognition Benefit from Auxiliary Information in Training?

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1 Supplemental Material for Can Visual Recognition Benefit from Auxiliary Information in Training? Qilin Zhang, Gang Hua, Wei Liu 2, Zicheng Liu 3, Zhengyou Zhang 3 Stevens Institute of Technolog, Hoboken, NJ 2 IBM Thomas J. Watson Research Center, Yorktown Heights, NY 3 Microsoft Research, Redmond, WA In this supplemental material, we present the proof of Proposition in Section and additional experimental results with the the RGBD Object dataset [] in Section 2. Proof of Proposition In this section, we present the proof of the Proposition for Paper 498. First, the existence of the upper bound is proven in Section., then the proof that the sequence f(w(s)), s =, 2, is monotonic is presented in Section.2. With the Bolzano-Weiersass theorem and the conclusions of Section. and Section.2, the Proposition is proven.. Proof of the existence of the upper bound C u From the consaint in Eq. (6) of the paper: [( τ j ) n ] R jr Tj τ j I n W j = I, () W T j where τ j denotes the pre-specified regularization parameter, 0 < τ j < (j =, 2,, J), we have ( ) ( τ j ) n WT j R j R T j W j τ j ( Wj T ) W j = n, (2) and therefore ) (W j T R j R T j W j In addition, we have ) (W j T R j R T k W k [ ) (W j T R j R T j W j 2 [ ] 2 τ j τ k τ j, 0 < τ j <, j =, 2,, J. (3) )] (W k T R k R T k W k (4) <, (5)

2 2 Q. Zhang, G. Hua, W. Liu, Z. Liu, Z. Zhang where the inequality (4) follows the property (B T A) 2 [ (A T A) (B T B) ], (6) which comes from the fact that ( (A B) T (A B) ) ) 0, because the maix (A B) T (A B) 0. Since (W j T R jr T k W k is bounded,, g(x) = x or x 2, we have ( )) g (W j T R j R T k W k [ ] 2 <. (7) 4 τ j τ k Considering c = 0 or, we have j k ( ( )) c g n WT j R j R T k W k 4n [ ] 2 <, (8) τ j τ k which shows that the sequence f(w(s)), s =, 2, is upper bounded by C u = 4n j k [ ] 2 <. (9) τ j τ k j k.2 Proof that the sequence f(w(s)), s =, 2, is monotonically increasing In this section, the monotonic property of the sequence f(w(s)), s =, 2, is presented. Following [2 4], we first present a Lemma, and then prove that f(w(s)) f(w(s )), s =, 2,, (0) where s is the iteration index, s =, 2,. Define the function r(y j, Y k ) def = ( n YT j Y k) = ( n WT j R jr T k W k ), therefore, f(w (s),, W J (s)) = j,,k j Using this notation, we have the following Lemma: Lemma. Define f j (W j ) def = j c g [ r(r T j W j, Y k (s )) ] s.t. W T j N j W j = I, c g[r(y j (s), Y k (s))]. () k=j c g [ r(r T j W j, Y k (s)) ] (2)

3 Visual Recognition with Auxiliary Information in Training 3 then f j (W j (s)) f j (W j (s )), j =,, J. (3) Proof. We prove Lemma in two cases, i.e., g(x) = x 2 and g(x) = x. Case : when g(x) = x 2, we have f j (W j ) in the following form, j ) 2 f j (W j ) = c (r(r T j W j, Y k (s )) which can be written as f j (W j (s)) = n where θ (s) k=j j (W j(s) T R j Y k (s )) n k=j = n j W j (s) T R j are defined as θ (s) = c (r(r T j W j, Y k (s))) 2, (4) (W j(s) T R j Y k (s)) (5) Y k(s ) k=j Y k(s), (6) { r (Y j (s), Y k (s )) if k =,, j r (Y j (s), Y k (s)) if k = j,, J. (7) ( j Note that in Eq. (6), the term c θ (s) Y k(s ) ) J k=j c θ (s) Y k(s) is equivalent to the definition of Z j (s), hence f j (W j ) can be simplified as n (WT j R jz j (s)). Considering the following optimization problem: whose solution is exactly we have max Wj n (WT j R j Z j (s)), s.t. W T j N j W j = I, (8) ( W j (s ) = N j R j Z j (s) [Z j (s) T R T j N j R j Z j (s)] /2), (9) (W j (s) T R j Z j (s)) (W j (s ) T R j Z j (s)). (20)

4 4 Q. Zhang, G. Hua, W. Liu, Z. Liu, Z. Zhang Similarly, the following equations can be obtained: j f j (W j (s)) = r(y j(s), Y k (s )) j r(y j(s ), Y k (s )) k=j k=j r(y j(s), Y k (s)) r(y j(s ), Y k (s)). (2) Considering that c is either 0 or, we hence have c = c 2. Applying the Cauchy-Schwartz inequality, we have j f j (W j (s)) c 2 θ (s) r(y j(s ), Y k (s )) j ( c θ (s) ) 2 k=j c ( ) 2 θ (s) j c (r(y j (s ), Y k (s ))) 2 = j c (r(y j (s), Y k (s ))) 2 j c (r(y j (s ), Y k (s ))) 2 = f j (W j (s)) k=j k=j k=j k=j c 2 θ (s) r(y j(s ), Y k (s)) c (r(y j (s ), Y k (s))) 2 c (r(y j (s), Y k (s))) 2 c (r(y j (s ), Y k (s))) 2 f j (W j (s )). (22) We immediately have f j (W j (s)) f j (W j (s )). This concludes the case scenario. Case 2: when g(x) = x, we have j f j (W j ) = c r(r T j W j, Y k (s )) k=j c r(r T j W j, Y k (s)). (23) Therefore, we can have exactly the same equation as Eq. (6), except that θ (s) for all the cases. The same equation as in Eq. (20) can be obtained, which directly implies that f j (W j (s)) f j (W j (s )). This concludes both the case 2 scenario and the entire proof of Lemma.

5 Visual Recognition with Auxiliary Information in Training 5 With the conclusion in Lemma, we proceed with the proof that the sequence f(w(s)), s =, 2, is monotonically increasing. Consider the following subaction [f j (W j (s )) f j (W j (s))] j= = j j= j= k=j j c g j= j j= = 2 [ c g n W j(s ) T R j R T k W k (s ) c g n W j(s ) T R j R T k W k (s) n W j(s) T R j R T k W k (s ) c g n W j(s ) T R j R T k W k (s ) j,,k j j,,k j c g [ ( ) ] n W j(s ) T R j R T k W k (s ) c g [ ( ) ] ] n W j(s) T R j R T k W k (s) 0. (24) The last equation in Eq. (24) follows the Lemma. This implies that f(w (s),, W J (s)) f(w (s ),, W J (s )) (25) i.e., f(w(s)) f(w(s )), s =, 2,. (26) Using Eq. (8), Eq. (26), the bounded sequence f(w(s)), s =, 2, is monotonically increasing. According to the Bolzano-Weiersass theorem, the sequence will converge, i.e., Proposition is proven. 2 Additional Experimental Results With the same RGBD Object dataset and experimental settings from [] and [5], we have also conducted additional experiments with the state-of-the-art HMPbased features [5]. These additional results are summarized in Table 2 (too many enies to fit in a single page). As is seen in Table 2, the overall recognition accuracy improves significantly across all methods, as compared to the EMK-based features []. In a large portion of the categories, perfect recognition is achieved even with the naive baseline

6 6 Q. Zhang, G. Hua, W. Liu, Z. Liu, Z. Zhang Table. Accuracy Table Part for the Multi-View RGBD Object Instance recognition with HMP features, the highest and second highest values are colored red and blue, respectively. The remaining part is in Table 2. Category SVM SVM2K KCCA KCCA RGCCA RGCCA RGCCA L L AL DCCA apple ball banana bell pepper binder bowl calculator camera cap cellphone cereal box coffee mug comb dry battery flashlight food bag food box food can food cup food jar garlic glue stick greens hand towel instant noodles keyboard kleenex lemon light bulb lime marker mushroom notebook onion orange peach pear pitcher plate pliers potato rubber eraser scissors shampoo soda can

7 Visual Recognition with Auxiliary Information in Training 7 Table 2. Continued from Table : Accuracy Table Part 2 for the Multi-View RGBD Object Instance recognition with HMP features, the highest and second highest values are colored red and blue, respectively. Category SVM SVM2K KCCA KCCAL RGCCA RGCCAL RGCCAAL DCCA sponge stapler tomato tooth brush tooth paste water bottle average SVM algorithm, However, the advantage of the proposed method is revealed in some of the challenging categories. Overall, with the HMP-based features, KCCAL and the RGCCAL algorithms cannot match the SVM baseline, and the SVM2K and KCCA algorithms are only marginally better than the baseline. The RGCCA and RGCCAAL algorithms offer some improvements, while the proposed DCCA algorithm achieves the highest overall recognition accuracy. Considering there are more than 3,000 testing samples, the two percent performance improvement means correctly classifying an additional amount of more than 200 samples. References. Lai, K., Bo, L., Ren, X., Fox, D.: A large-scale hierarchical multi-view rgb-d object dataset. In: Robotics and Automation (ICRA), 20 IEEE International Conference on, IEEE (20) Tenenhaus, A., Tenenhaus, M.: Regularized generalized canonical correlation analysis. Psychomeika 76 (20) Tenenhaus, A.: Kernel generalized canonical correlation analysis. Actes des 42è journées de Statistique (200) 4. Hanafi, M.: Pls path modelling: computation of latent variables with the estimation mode b. Computational Statistics 22 (2007) Bo, L., Ren, X., Fox, D.: Unsupervised feature learning for rgb-d based object recognition. ISER, June (202)