Richard Sinkhorn. relationship between arbitrary positive matrices doubly stochastic matrices. Annals Mathematical Statistics, 35(2):876--879, June 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The method presented here works with the edge image of local features (which gives an approximate notion of the 2D shape of that feature) rather than their intensity. A doubly stochastic matrix, representing the probability of match between the features, is obtained using Sinkhorn normalization [7] and the prior information. A statistically optimal technique is proposed which relies on minimizing the probability of error of a mismatch or equivalently # Partially supported by NSF ITR grant #0086075 maximizing the posterior density of the match given a feature. The method works by matching ....

....the probability of matching the The notation X i (n) represents the i th feature from the n th viewing position. elements of X given a particular Y. This is done by using the Sinkhorn normalization procedure to obtain a doubly stochastic matrix by alternating row and column normalizations [7]. This allows us to use either X or Y as the reference feature set. 2.5. The Global Matching Scheme Rather than computing a probability of match for individual features, a more reliable correspondence can be obtained if we consider the entire set of features, taking into account their spatial ....

R. Sinkhorn, "A relationship between arbitrary positive matrices and doubly stochastic matrices," Annals Math. Statist., vol. 35, pp. 876--879, 1964.

.... taken to be me average of the distances between each point and its closest neighbor (which we denote os) For simplicity of analysis, we normalize the affinity matrix by projecting it to the nearest doubly stochastic matrix P diag(d)Adiag(d) using a fast modification of the Sinkhorn procedure [28] to solve for d satisfying PI 1 (and pT 1 1 since pT p) While not crucial to this inethod, a doubly stochastic matrix has two properties that make it appealing as a model of the data: P can be interpreted as the transition probabilities of a random walk on the data; and P s largest ....

R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of.Mathemat- ical Statistics, 35:876-879, 1964.

....The method presented here works with the edge image of local features (which gives an approx imate notion of the 2D shape of that feature) rather than their intensity. A doubly stochastic matrix, representing the probability of match between the features, is obtained using Sinkhorn normalization [16] and the prior information. A statistically optimal technique is proposed, which relies on minimizing the probability of error of a mismatch or equivalently maximizing the pos terior density of the match given one of the features. The method works by matching the entire constellation of features ....

....the probability of matching the elements of Y given a particular X, and each column the probability of matching the elements of X given a particular Y. This is done by using the Sinkhorn normalization procedure to obtain a doubly stochastic matrix by alternating row and column normalizations [16]. The advantage of using the Sinkhorn normalization procedure is that it allows us to use either X or Y as the reference feature set. It requires a priori identification of unpaired features. This reduces the number of features that need to be matched and hence the combinatorics of the problem. ....

R. Sinkhorn, "A relationship between arbitrary positive matrices and doubly stochastic matrices," Annals Math. Statist., vol. 35, pp. 876-879, 1964.

....peaks are evident. c) The same example after normalization. To solve this problem, a set of weights w ij must be assigned to the epipolar lines in that deemphasizes features with many possible matches. These weights must also satisfy Eq. 10) We use an iterative normalization procedure [Sin64] to transform the initial (invalid) match matrix into a (valid) doubly stochastic matrix. First, the matrix W is set to zero; entries for matches satisfying the geometric constraints of Section 4.2.2, as well as all entries in row M 1 and column N 1, are then assigned an initial value of one. ....

Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876--879, June 1964.

....problem. See Yuille and Rangarajan, in preparation, for more details) Our nal example is a discrete iterative algorithm to solve the linear assignment problem. This algorithm was reported by Kosowsky and Yuille in [5] where it was also shown to correspond to the well known Sinkhorn algorithm [9]. We now show that both Kosowsky and Yuille s linear assignment algorithm, and hence Sinkhorn s algorithm are examples of CCCP (after a change of variables) Example 3. The linear assignment problem seeks to nd the permutation matrix ia g which minimizes the energy E[ ia A ia , ....

R. Sinkhorn. \A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices". Ann. Math. Statist.. 35, pp 876-879. 1964.

....weighted (Figure 6 8) solve problem, a mutually consistent set weights w must assigned the epipo lines M such that features having many possible matches de emphasized. order ensure that the normalization condition in (6 10) is satisfied, iterative normalization procedure proposed by Sinkhorn [Sin64, Sin67] is utilized transform initial (invalid) match matrix into valid doubly stochastic matrix. First, the matrix to zero; entries for matches satisfying the geometric constraints 6.1.2, well entries row M and column N , are then assigned initial value one. Sinkhorn s algorithm alternatively ....

....peaks are evident. same example after normalization. follows: # ij ij # w # 1, M ## ij # ij # w # # 1, N . Each entry the matrix normalized sum entries row; each entry the resulting matrix then normalized entries column, and on. The algorithm produces provably unique factorization WD [Sin64], such that doubly stochastic. The new matrix does represent correct distribution, because it is somewhat arbitrarily initial ized, provides practical approximation purposes Hough transform technique described above. Recall from 4.1.5 planar accumulation space, each linear constraint form (6 1) ....

Richard Sinkhorn. relationship between arbitrary positive matrices doubly stochastic matrices. Annals Mathematical Statistics, 35(2):876--879, June 1964.

....is by generalizing work by Kosowsky and Yuille [7] 8] who used a result similar to Theorem 5 to obtain an algorithm for solving the linear assignment problem. Kosowsky and Yuille [7] 8] also showed that this result could be used to derive an energy function for the classic Sinkhorn algorithm [17] which converts positive matrices into doubly stochastic ones. Rangarajan et al. [13] applied this result to obtain double loop algorithms for a range of optimization problems subject to linear constraints. In the next section we apply Theorems 4 and 5 to the Bethe free energy. In particular, we ....

R. Sinkhorn. \A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices". Ann. Math. Statist.. 35, pp 876-879. 1964.

....To solve this problem, a mutually consistent set of weights w ij must be assigned to the epipolar lines in M such that features having many possible matches are de emphasized. In order to ensure that the condition in (14) is satisfied, an iterative normalization procedure proposed by Sinkhorn [47, 16] is utilized to transform an initial (invalid) match matrix into a valid doubly stochastic matrix. First, the matrix W is set to zero; entries for matches satisfying the geometric constraints of 3.2.2, as well as all entries in row M 1 and column N 1, are then assigned an initial value of ....

....= w # ij # M # i=1 w # ij #j # 1, N . Each entry in the matrix is normalized by the sum of entries in its row; each entry in the resulting matrix is then normalized by the sum of entries in its column, and so on. The algorithm produces a provably unique factorization W # = D 1 WD 2 [47], such that W # is doubly stochastic. The new matrix does not represent the correct distribution, because it is somewhat arbitrarily initialized, but it provides a useful approximation for the purposes of the Hough transform technique described above. For a planar accumulation space, each linear ....

Richard Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876--879, June 1964.

....a threshold, determining when to select no match (slack assignment) over a match with distance d jk . See [11] for an analytical justification. The continuous matrix M 0 converges toward the discrete matrix M due to two mechanisms that are used concurrently: 1. First, a technique due to Sinkhorn [19] is applied. When each row and column of a square correspondence matrix is normalized (several times, alternatingly) by the sum of the elements of that row or column respectively, the resulting matrix has positive elements with all rows and columns summing to one. 2. The term fi is increased as ....

Sinkhorn, R., "A Relationship between Arbitrary Positive Matrices and Doubly Stochastic Matrices", Annals Math. Statist., 35, pp. 876--879, 1964.

....to 0. Second, a 1 is placed in the matrix for each plausible match (using the criteria of Section 4.2) Third, a 1 is placed in every outlier row and column. Finally, the matrix is reduced to doubly stochastic form, so that all rows and columns sum to unity, by application of Sinkhorn s algorithm [21, 19]. Weighting each line by its approximate likelihood dramatically improves the coherence of HT peaks. After all epipolar lines have been drawn in the HT, a set of candidate peaks is found by searching the Hough image for relative maxima, i.e. points whose value exceeds all others in a square ....

Sinkhorn, R. "A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices". Annals of Mathematical Statistics, Vol. 35, No. 2, June 1964, pp. 876-879.

....1] or by failing to satisfy the row and column constraints. The procedure applied to enforce the row and column constraints involves renormalizing the activities after each harmony update to bring the activity pattern arbitrarily close to a doubly stochastic matrix. The procedure, suggested by Sinkhorn (1964), involves alternating row and column normalizations (in our case to the values of the mask vector) Sinkhorn proved that this procedure will asymptotically converge on a doubly stochastic matrix. Note that the Sinkhorn normalization procedure must operate at a much finer time grain than the ....

Sinkhorn, Richard (1964). A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices. Annals of Mathematical Statistics, Vol. 35, No. 2. pp. 876-879.

....the row and column constraints (Equations 1) The procedure applied to enforce the row and column constraints involves renormalizing the activities after each harmony update (Equation 3) to bring the activity pattern arbitrarily close to a doubly stochastic matrix. The procedure, suggested by Sinkhorn (1964), involves alternating row and column normalizations (in our case to the values of the mask vector) Sinkhorn proved that this procedure will asymptotically converge on a doubly stochastic matrix. Note that the Sinkhorn normalization procedure must operate at a much ner time grain than the ....

Sinkhorn, Richard (1964). A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices. Annals of Mathematical Statistics, Vol. 35, No. 2. pp. 876-879.

....Therefore, the number of iterations is limited to t max , which is set at 100 in the experiments in Section 4. When constraint (13d) and hence step 4(c) is removed, the above algorithm is almost equivalent to the soft assign ( DCN) In this case, there are convergence proofs for step 4 (Sinkhorn, 1964) 3 , and for the whole basic algorithm (Rangarajan, Yuille, Gold and Mjolsness, 1997) With step 4(c) however, the convergence of step 4 has not been proven yet. Nevertheless, our computer simulations imply that step 4, including step 4(c) always converges. The effect of step 4(c) will be ....

Sinkhorn, R. (1964). A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35, 876-879.

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Richard Sinkhorn. relationship between arbitrary positive matrices doubly stochastic matrices. Annals Mathematical Statistics, 35(2):876--879, June 1964.

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R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35:876--879, 1964.

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R. Sinkhorn, "A relationship between arbitrary positive matrices and doubly stochastic matrices," Ann. Math. Statist., vol. 35, pp. 876--879, 1964.

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R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. The Annals of Mathematical Statistics, 35(2):876--879, 1964.

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R. Sinkhorn (1964), A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35, pp. 876--879. 19

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R. Sinkhorn. "A Relationship between Arbitrary Positive Matrices and Doubly Stochastic Matrices", Annals Math. Statist., vol. 35, pp. 876--879, 1964.

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R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. The Annals of Mathematical Statistics, 35(2):876--879, 1964.

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R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35:876--879, 1964.

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R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35:876--879, 1964.

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Sinkhorn, R. 1964. A Relationship between Arbitrary Positive Matrices and Doubly Stochastic Matrices. Annals Mathematical Statistics, 35(2):876--879.

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R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35, 876-879, 1964.

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R. Sinkhorn, "A Relationship between Arbitrary Positive Matrices and Doubly Stochastic Matrices ", Annals Math. Statist., vol. 35, pp. 876--879, 1964.

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