Face recognition and ranking data - University of Illinois Chicago
Face Recognition and Ranking Data
1 Face Recognition problem and dimension machine learning covariances and eigenfaces
2 Ranking Data engineering a search engine the random surfer model beyond PageRank
MCS 472 Lecture 25 Industrial Math & Computation Jan Verschelde, 8 March 2023
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
1 / 30
Face Recognition and Ranking Data
1 Face Recognition problem and dimension machine learning covariances and eigenfaces
2 Ranking Data engineering a search engine the random surfer model beyond PageRank
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
2 / 30
face recognition
Face recognition is challenging because of 1 the position of the head, 2 lightning conditions, and 3 moods and expressions.
Many automated systems have been developed. A linear algebra approach is based on eigenfaces, a method proposed by Sirovich and Kirby, 1987.
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
3 / 30
dimension reduction
The input is a p-by-q grayscale image. The resolution of an image is m = p ? q.
All images of the same resolution live in an m-dimensional space. The subspace of all facial images has a low dimension,
independent of the resolution.
This result is described in the paper by Neil Muller, Louren?o Magaia, B. M. Herbst: Singular Value Decomposition, Eigenfaces, and 3D Reconstructions. SIAM Review, Vol. 46, No. 3, pages 518?545, 2004.
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
4 / 30
the singular value decomposition
The singular value decomposition of a p-by-q matrix A is A = UV T , U Rp?p, Rp?q, V Rq?q,
where U and V are orthogonal: U-1 = UT , V -1 = V T , and is a diagonal matrix, with on its diagonal
1 2 ? ? ? min(p,q), are the singular values of the matrix A. If rank(A) = r , then i = 0 for all i > r . Ignoring the smallest singular values leads to a dimension reduction.
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
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the geometric interpretation of A = UV T
1 rotate a circle
v2
v1
2 stretch the circle 3 rotate the ellipse
v2 v1
v2 v1
v2 v2 u2
v1
v1 u1
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
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relationship with eigenvalues and eigenvectors
By the singular value decomposition of A: A = UV T AV = U = U.
This implies
Avj = uj , j = 1, 2, . . . , min(p, q).
The largest singular value measures the magnitude of A: A2 = max Ax2 = 1.
x 2 =1
Exercise 1: Verify that A = UV T implies AT AV = V T and AAT U = UT .
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
7 / 30
Face Recognition and Ranking Data
1 Face Recognition problem and dimension machine learning covariances and eigenfaces
2 Ranking Data engineering a search engine the random surfer model beyond PageRank
Industrial Math & Computation (MCS 472)
face recognition and ranking data
L-25 8 March 2023
8 / 30
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