We continue discussing Convolutional Neural Network (CNN) in image and object embedding in a shared space, shape signature and image retrieval.
We were discussing the Euclidean distance and the distance matrix. As we know with Euclidean distance, the chi-square measure, which is the sum of squares of errors, gives a good indication of how close the objects are to the mean. Therefore it is a measure for the goodness of fit. The principle is equally applicable to embedding space. By using a notion of errors, we can make sure that the shapes and images embedded in the space do not violate the intra member distances. The only variation the authors applied to this measure is the use of Sammon error instead of the chi-square because it encourages the preservation of the structure of local neighborhoods while embedding. The joing embedding space is a Euclidean space of lower dimension while the shapes and the images are represented in the original high dimensional space.
The Sammon Error is a weighted sum of differences between the original pairwise distances and the embedding pairwise distances. Dissimilar shapes have more differences and therefore they are weighted down.
The embedding of shapes and images proceeds with minimizing the Sammon Error using non-linear Multi-dimensional scaling. It is a means of visualizing the level of similarity of individual cases in a dataset. By minimizing the intra member distance in the placement of items in an N-dimensional space. The goal of MDS is to get the co-ordinate matrix. It uses the notion that the co-ordinate matrix can easily be formed from the eigenvalue decomposition of the scalar product matrix B. The steps of a classical MDS algorithm are as follows:
1. Form pairwise distance matrix that gives proximity between pairs
2. Multiply the squared proximities with a form of identity matrix to get matrix B
3. Extract the m-largest positive eigenvalues and the corresponding m eigenvectors
4. Form the co-ordinate matrix from the eigen vectors and the diagonal matrix of the eigen values
#proposal for login screens ui enhancements: https://1drv.ms/w/s!Ashlm-Nw-wnWtWrw5g03hYU5CBKL
We were discussing the Euclidean distance and the distance matrix. As we know with Euclidean distance, the chi-square measure, which is the sum of squares of errors, gives a good indication of how close the objects are to the mean. Therefore it is a measure for the goodness of fit. The principle is equally applicable to embedding space. By using a notion of errors, we can make sure that the shapes and images embedded in the space do not violate the intra member distances. The only variation the authors applied to this measure is the use of Sammon error instead of the chi-square because it encourages the preservation of the structure of local neighborhoods while embedding. The joing embedding space is a Euclidean space of lower dimension while the shapes and the images are represented in the original high dimensional space.
The Sammon Error is a weighted sum of differences between the original pairwise distances and the embedding pairwise distances. Dissimilar shapes have more differences and therefore they are weighted down.
The embedding of shapes and images proceeds with minimizing the Sammon Error using non-linear Multi-dimensional scaling. It is a means of visualizing the level of similarity of individual cases in a dataset. By minimizing the intra member distance in the placement of items in an N-dimensional space. The goal of MDS is to get the co-ordinate matrix. It uses the notion that the co-ordinate matrix can easily be formed from the eigenvalue decomposition of the scalar product matrix B. The steps of a classical MDS algorithm are as follows:
1. Form pairwise distance matrix that gives proximity between pairs
2. Multiply the squared proximities with a form of identity matrix to get matrix B
3. Extract the m-largest positive eigenvalues and the corresponding m eigenvectors
4. Form the co-ordinate matrix from the eigen vectors and the diagonal matrix of the eigen values
#proposal for login screens ui enhancements: https://1drv.ms/w/s!Ashlm-Nw-wnWtWrw5g03hYU5CBKL
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