Semidefinite embedding

Maximum Variance Unfolding (MVU), also known as Semidefinite Embedding (SDE), is an algorithm in computer science that uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data.^[1][2][3]

It is motivated by the observation that kernel Principal Component Analysis (kPCA) does not reduce the data dimensionality,^[4] as it leverages the Kernel trick to non-linearly map the original data into an inner-product space.

Algorithm

MVU creates a mapping from the high dimensional input vectors to some low dimensional Euclidean vector space in the following steps:^[5]

1. A neighbourhood graph is created. Each input is connected with its k-nearest input vectors (according to Euclidean distance metric) and all k-nearest neighbors are connected with each other. If the data is sampled well enough, the resulting graph is a discrete approximation of the underlying manifold.
2. The neighbourhood graph is "unfolded" with the help of semidefinite programming. Instead of learning the output vectors directly, the semidefinite programming aims to find an inner product matrix that maximizes the pairwise distances between any two inputs that are not connected in the neighbourhood graph while preserving the nearest neighbors distances.
3. The low-dimensional embedding is finally obtained by application of multidimensional scaling on the learned inner product matrix.

The steps of applying semidefinite programming followed by a linear dimensionality reduction step to recover a low-dimensional embedding into a Euclidean space were first proposed by Linial, London, and Rabinovich.^[6]

Optimization formulation

Let ${\displaystyle X\,\!}$ be the original input and ${\displaystyle Y\,\!}$ be the embedding. If ${\displaystyle i,j\,\!}$ are two neighbors, then the local isometry constraint that needs to be satisfied is:^[7][8][9]

${\displaystyle |X_{i}-X_{j}|^{2}=|Y_{i}-Y_{j}|^{2}\,\!}$

Let ${\displaystyle G,K\,\!}$ be the Gram matrices of ${\displaystyle X\,\!}$ and ${\displaystyle Y\,\!}$ (i.e.: ${\displaystyle G_{ij}=X_{i}\cdot X_{j},K_{ij}=Y_{i}\cdot Y_{j}\,\!}$). We can express the above constraint for every neighbor points ${\displaystyle i,j\,\!}$ in term of ${\displaystyle G,K\,\!}$:^[10][11]

${\displaystyle G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji}\,\!}$

In addition, we also want to constrain the embedding ${\displaystyle Y\,\!}$ to center at the origin:^[12][13][14]

${\displaystyle 0=|\sum _{i}Y_{i}|^{2}\Leftrightarrow (\sum _{i}Y_{i})\cdot (\sum _{i}Y_{i})\Leftrightarrow \sum _{i,j}Y_{i}\cdot Y_{j}\Leftrightarrow \sum _{i,j}K_{ij}}$

As described above, except the distances of neighbor points are preserved, the algorithm aims to maximize the pairwise distance of every pair of points. The objective function to be maximized is:^[15][16][17]

${\displaystyle T(Y)={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}}$

Intuitively, maximizing the function above is equivalent to pulling the points as far away from each other as possible and therefore "unfold" the manifold. The local isometry constraint ^[18]

Let ${\displaystyle \tau =max\{\eta _{ij}|Y_{i}-Y_{j}|^{2}\}\,\!}$ where ${\displaystyle \eta _{ij}:={\begin{cases}1&{\mbox{if}}\ i{\mbox{ is a neighbour of }}j\\0&{\mbox{otherwise}}.\end{cases}}}$

prevents the objective function from diverging (going to infinity).

Since the graph has N points, the distance between any two points ${\displaystyle |Y_{i}-Y_{j}|^{2}\leq N\tau \,\!}$. We can then bound the objective function as follows:^[19][20]

${\displaystyle T(Y)={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}\leq {\dfrac {1}{2N}}\sum _{i,j}(N\tau )^{2}={\dfrac {N^{3}\tau ^{2}}{2}}\,\!}$

The objective function can be rewritten purely in the form of the Gram matrix:^[21][22][23]

${\displaystyle {\begin{aligned}T(Y)&{}={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}\\&{}={\dfrac {1}{2N}}\sum _{i,j}(Y_{i}^{2}+Y_{j}^{2}-Y_{i}\cdot Y_{j}-Y_{j}\cdot Y_{i})\\&{}={\dfrac {1}{2N}}(\sum _{i,j}Y_{i}^{2}+\sum _{i,j}Y_{j}^{2}-\sum _{i,j}Y_{i}\cdot Y_{j}-\sum _{i,j}Y_{j}\cdot Y_{i})\\&{}={\dfrac {1}{2N}}(\sum _{i,j}Y_{i}^{2}+\sum _{i,j}Y_{j}^{2}-0-0)\\&{}={\dfrac {1}{N}}(\sum _{i}Y_{i}^{2})={\dfrac {1}{N}}(Tr(K))\\\end{aligned}}\,\!}$

Finally, the optimization can be formulated as:^[24][25][26]

${\displaystyle {\begin{aligned}&{\text{Maximize}}&&Tr(\mathbf {K} )\\&{\text{subject to}}&&\mathbf {K} \succeq 0,\sum _{ij}\mathbf {K} _{ij}=0\\&{\text{and}}&&G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji},\forall i,j{\mbox{ where }}\eta _{ij}=1,\end{aligned}}}$

After the Gram matrix ${\displaystyle K\,\!}$ is learned by semidefinite programming, the output ${\displaystyle Y\,\!}$ can be obtained via Cholesky decomposition.

In particular, the Gram matrix can be written as ${\displaystyle K_{ij}=\sum _{\alpha =1}^{N}(\lambda _{\alpha }V_{\alpha i}V_{\alpha j})\,\!}$ where ${\displaystyle V_{\alpha i}\,\!}$ is the i-th element of eigenvector ${\displaystyle V_{\alpha }\,\!}$ of the eigenvalue ${\displaystyle \lambda _{\alpha }\,\!}$.^[27][28]

It follows that the ${\displaystyle \alpha \,\!}$-th element of the output ${\displaystyle Y_{i}\,\!}$ is ${\displaystyle {\sqrt {\lambda _{\alpha }}}V_{\alpha i}\,\!}$.^[29][30]

See also

• Locally linear embedding
• Isometry (disambiguation)
• Local Tangent Space Alignment
• Riemannian manifold
• Energy minimization

Notes

1. Weinberger, Sha and Saul 2004a
2. Weinberger and Saul 2004b
3. Weinberger and Saul 2006
4. Lawrence 2012, page 1612
5. Weinberger, Sha and Saul 2004a, page 7.
6. Linial, London and Rabinovich 1995
7. Weinberger, Sha and Saul 2004a, page 3, equation 8
8. Weinberger and Saul 2004b, page 3, equation 2
9. Weinberger and Saul 2006, page 4, equation 2
10. Weinberger, Sha and Saul 2004a, page 3, equation 9
11. Weinberger and Saul 2004b, page 3, equation 3
12. Weinberger, Sha and Saul 2004a, page 3, equation 6
13. Weinberger and Saul 2004b, page 3, equation 5
14. Weinberger and Saul 2006, page 5, equation 8
15. Weinberger, Sha and Saul 2004a, page 4, equation 10
16. Weinberger and Saul 2004b, page 4, equation 6
17. Weinberger and Saul 2006, page 5, equation 4
18. Weinberger and Saul 2004b, page 4, equation 7
19. Weinberger and Saul 2004b, page 4, equation 8
20. Weinberger and Saul 2006, page 5, equation 6
21. Weinberger, Sha and Saul 2004a, page 4, equation 11
22. Weinberger and Saul 2004b, page 4, equation 9
23. Weinberger and Saul 2006, page 6, equations 10 to 13
24. Weinberger, Sha and Saul 2004a, page 4, section 3.3
25. Weinberger and Saul 2004b, page 4, equation 9
26. Weinberger and Saul 2006, page 6, equations 10 to 13
27. Weinberger and Saul 2004b, page 4, equation 10
28. Weinberger and Saul 2006, page 7, equations 14
29. Weinberger and Saul 2004b, page 4, equation 11
30. Weinberger and Saul 2006, page 7, equations 15

References

• Linial, London and Rabinovich, Nathan, Eran and Yuri (1995). "The geometry of graphs and some of its algorithmic applications". Combinatorica. 15 (2): 215–245. doi:10.1007/BF01200757. S2CID 5071936.
• Weinberger, Sha and Saul, Kilian Q., Fei and Lawrence K. (4 July 2004a). Learning a kernel matrix for nonlinear dimensionality reduction. Proceedings of the Twenty First International Conference on Machine Learning (ICML 2004). Banff, Alberta, Canada.
• Weinberger and Saul, Kilian Q. and Lawrence K. (27 June 2004b). Unsupervised learning of image manifolds by semidefinite programming. 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Vol. 2.
• Weinberger and Saul, Kilian Q. and Lawrence K. (1 May 2006). "Unsupervised learning of image manifolds by semidefinite programming" (PDF). International Journal of Computer Vision. 70: 77–90. doi:10.1007/s11263-005-4939-z. S2CID 291166.
• Lawrence, Neil D (2012). "A unifying probabilistic perspective for spectral dimensionality reduction: insights and new models". Journal of Machine Learning Research. 13 (May): 1612. arXiv:1010.4830. Bibcode:2010arXiv1010.4830L.

Additional material

• Kilian Q. Weinberger's MVU Matlab code