README.md

SPD_reduce

This repo is dedicated to Deep Manifold Learning of Symmetric Positive Definite (SPD) Matrices with Application to Face Recognition article from AAAI 2017.

The main idea is too apply article ideas about SPD matrixes manifold to real world problems via finding low-level representation of SPD matrixes. In this project next thee problems we're considered:

• Encoding random SPD matrixes using basic blocks from article
• Speed up determinant calculation for SPD matrixes via this encoding
• Use basic blocks to solve problem of finding embeddings for graph (MUTAG dataset is used for this task)

Dependencies:

• pytorch (v. 0.4.0)
• networkx (v. 2.0)
• sklearn (v. 0.19.1)
• tqdm (v. 4.19.7+)

Encoding SPD matrixes

For that task two types of basic block are used:

• 2D Fully Connected Layer (SPD.Linear2D class)
• Symmetrically Clean Layer (SPD.SymmetricallyCleanLayer class)

With this block's we constract model for matrix encoding (SPD.MatrixEncoder class) with next architecture:

MatrixEncoder(
  (encoder): Sequential(
    (0): Linear2D()
    (1): SymmetricallyCleanLayer(
      (relu): ReLU()
    )
    (2): Tanh()
  )
  (decoder): Sequential(
    (0): Linear2D()
    (1): ReLU()
  )
)

Using sklearn.datasets.make_spd_matrix we conctract dataset of SPD matrixes and train MatrixEncoder with simple MSELoss and Adam optimizer.

Code examples

run test using model based on basic blocks

from SPD import run_test, run_test_vae, run_test_conv
coder, dataset = run_test(20) 

based on VAE AutoEncoder

coder, dataset = run_test_vae(20) 

based on Convolution AutoEncoder

coder, dataset = run_test_conv(20)

Determinant calculation

Using encoder model from previous task we would like to do next: using this encoder we embed matrix A in low-dimensional matrix B and using matrix B we predict determinant of matrix A with another model.

Architecture for predictor is simple feed-forward network with 3 layers:

DetNet(
  (fc1): Linear(in_features=100, out_features=64)
  (fc2): Linear(in_features=64, out_features=32)
  (fc3): Linear(in_features=32, out_features=1)
)

It's important to mention, that MatrixEncoder and DetNet are trained separately.

Code examples

from SPD import run_detetmenant
coder, dataset = run_detetmenant(20) 

Embeddings for MUTAG dataset

Using MatrixEncoder model we'd like to constract embeddings for graph structures and solve clustering, classification, anomaly detection problems and etc.

MUTAG dataset is used as benchmark in this task (~200 graphs with two labels).

Code examples

run test using model based on basic blocks

from SPD import run_test_mutag, run_test_vae_mutag
coder, dataset = run_test_mutag(20) 

based on VAE AutoEncoder

coder, dataset = run_test_vae_mutag(20) 

References:

• Deep Manifold Learning of Symmetric Positive Definite (SPD) Matrices with Application to Face Recognition
• Visualizing Data using t-SNE
• Hessian Eigenmaps: new locally linear embedding techniques for high-dimensional data
• Adaptive Manifold Learning
• Learning with kernels
• A Global Geometric Frameworkfor Nonlinear DimensionalityReduction