Dimension reduction with OT
Warning
Note that by default the module is not imported in ot. In order to use it you need to explicitly import ot.dr
Compute squared euclidean distance between samples (autograd)
Entropic Wasserstein Component Analysis [52].
The function solves the following optimization problem:
\[\mathbf{U} = \mathop{\arg \min}_\mathbf{U} \quad W(\mathbf{X}, \mathbf{U}\mathbf{U}^T \mathbf{X})\]
where :
\(\mathbf{U}\) is a matrix in the Stiefel(p, d) manifold
\(W\) is entropic regularized Wasserstein distances
\(\mathbf{X}\) are samples
X (ndarray, shape (n, d)) – Samples from measure \(\mu\).
U0 (ndarray, shape (d, k), optional) – Initial starting point for projection.
reg (float, optional) – Regularization term >0 (entropic regularization).
k (int, optional) – Subspace dimension.
method (str, optional) – Eather ‘BCD’ or ‘MM’ (Block Coordinate Descent or Majorization-Minimization). Prefer MM when d is large.
sinkhorn_method (str) – Method used for the Sinkhorn solver, see ot.bregman.sinkhorn for more details.
stopThr (float, optional) – Stop threshold on error (>0).
maxiter (int, optional) – Maximum number of iterations of the BCD/MM.
maxiter_sink (int, optional) – Maximum number of iterations of the Sinkhorn solver.
maxiter_MM (int, optional) – Maximum number of iterations of the MM (only used when method=’MM’).
verbose (int, optional) – Print information along iterations.
pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters.
U (ndarray, shape (d, k)) – Matrix Stiefel manifold.
References
ot.dr.ewca
Fisher Discriminant Analysis
P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters
proj (callable) – projection function including mean centering
ot.dr.fda
Log-sum-exp reduction compatible with autograd (no numpy implementation)
Projection Robust Wasserstein Distance [32]
The function solves the following optimization problem:
\[\max_{U \in St(d, k)} \ \min_{\pi \in \Pi(\mu,\nu)} \quad \sum_{i,j} \pi_{i,j} \|U^T(\mathbf{x}_i - \mathbf{y}_j)\|^2 - \mathrm{reg} \cdot H(\pi)\]
\(U\) is a linear projection operator in the Stiefel(d, k) manifold
\(H(\pi)\) is entropy regularizer
\(\mathbf{x}_i\), \(\mathbf{y}_j\) are samples of measures \(\mu\) and \(\nu\) respectively
X (ndarray, shape (n, d)) – Samples from measure \(\mu\)
Y (ndarray, shape (n, d)) – Samples from measure \(\nu\)
a (ndarray, shape (n, )) – weights for measure \(\mu\)
b (ndarray, shape (n, )) – weights for measure \(\nu\)
tau (float) – stepsize for Riemannian Gradient Descent
U0 (ndarray, shape (d, p)) – Initial starting point for projection.
reg (float, optional) – Regularization term >0 (entropic regularization)
k (int) – Subspace dimension
stopThr (float, optional) – Stop threshold on error (>0)
verbose (int, optional) – Print information along iterations.
random_state (int, RandomState instance or None, default=None) – Determines random number generation for initial value of projection operator when U0 is not given.
pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters
U (ndarray, shape (d, k)) – Projection operator.
References
Sinkhorn algorithm with fixed number of iteration (autograd)
Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)
split samples in \(\mathbf{X}\) by classes in \(\mathbf{y}\)
Wasserstein Discriminant Analysis [11]
The function solves the following optimization problem:
\[\mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad \frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}\]
where :
\(P\) is a linear projection operator in the Stiefel(p, d) manifold
\(W\) is entropic regularized Wasserstein distances
\(\mathbf{X}^i\) are samples in the dataset corresponding to class i
Choosing a Sinkhorn solver
By default and when using a regularization parameter that is not too small the default sinkhorn solver should be enough. If you need to use a small regularization to get sparse cost matrices, you should use the ot.dr.sinkhorn_log() solver that will avoid numerical errors, but can be slow in practice.
X (ndarray, shape (n, d)) – Training samples.
y (ndarray, shape (n,)) – Labels for training samples.
p (int, optional) – Size of dimensionality reduction.
reg (float, optional) – Regularization term >0 (entropic regularization)
solver (None | str, optional) – None for steepest descent or ‘TrustRegions’ for trust regions algorithm else should be a pymanopt.solvers
sinkhorn_method (str) – method used for the Sinkhorn solver, either ‘sinkhorn’ or ‘sinkhorn_log’
P0 (ndarray, shape (d, p)) – Initial starting point for projection.
normalize (bool, optional) – Normalize the Wasserstaiun distance by the average distance on P0 (default : False)
verbose (int, optional) – Print information along iterations.
P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters
proj (callable) – Projection function including mean centering.
References
ot.dr.wda
Compute squared euclidean distance between samples (autograd)
Entropic Wasserstein Component Analysis [52].
The function solves the following optimization problem:
\[\mathbf{U} = \mathop{\arg \min}_\mathbf{U} \quad W(\mathbf{X}, \mathbf{U}\mathbf{U}^T \mathbf{X})\]
where :
\(\mathbf{U}\) is a matrix in the Stiefel(p, d) manifold
\(W\) is entropic regularized Wasserstein distances
\(\mathbf{X}\) are samples
X (ndarray, shape (n, d)) – Samples from measure \(\mu\).
U0 (ndarray, shape (d, k), optional) – Initial starting point for projection.
reg (float, optional) – Regularization term >0 (entropic regularization).
k (int, optional) – Subspace dimension.
method (str, optional) – Eather ‘BCD’ or ‘MM’ (Block Coordinate Descent or Majorization-Minimization). Prefer MM when d is large.
sinkhorn_method (str) – Method used for the Sinkhorn solver, see ot.bregman.sinkhorn for more details.
stopThr (float, optional) – Stop threshold on error (>0).
maxiter (int, optional) – Maximum number of iterations of the BCD/MM.
maxiter_sink (int, optional) – Maximum number of iterations of the Sinkhorn solver.
maxiter_MM (int, optional) – Maximum number of iterations of the MM (only used when method=’MM’).
verbose (int, optional) – Print information along iterations.
pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters.
U (ndarray, shape (d, k)) – Matrix Stiefel manifold.
References
Fisher Discriminant Analysis
P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters
proj (callable) – projection function including mean centering
Log-sum-exp reduction compatible with autograd (no numpy implementation)
Projection Robust Wasserstein Distance [32]
The function solves the following optimization problem:
\[\max_{U \in St(d, k)} \ \min_{\pi \in \Pi(\mu,\nu)} \quad \sum_{i,j} \pi_{i,j} \|U^T(\mathbf{x}_i - \mathbf{y}_j)\|^2 - \mathrm{reg} \cdot H(\pi)\]
\(U\) is a linear projection operator in the Stiefel(d, k) manifold
\(H(\pi)\) is entropy regularizer
\(\mathbf{x}_i\), \(\mathbf{y}_j\) are samples of measures \(\mu\) and \(\nu\) respectively
X (ndarray, shape (n, d)) – Samples from measure \(\mu\)
Y (ndarray, shape (n, d)) – Samples from measure \(\nu\)
a (ndarray, shape (n, )) – weights for measure \(\mu\)
b (ndarray, shape (n, )) – weights for measure \(\nu\)
tau (float) – stepsize for Riemannian Gradient Descent
U0 (ndarray, shape (d, p)) – Initial starting point for projection.
reg (float, optional) – Regularization term >0 (entropic regularization)
k (int) – Subspace dimension
stopThr (float, optional) – Stop threshold on error (>0)
verbose (int, optional) – Print information along iterations.
random_state (int, RandomState instance or None, default=None) – Determines random number generation for initial value of projection operator when U0 is not given.
pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters
U (ndarray, shape (d, k)) – Projection operator.
References
Sinkhorn algorithm with fixed number of iteration (autograd)
Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)
split samples in \(\mathbf{X}\) by classes in \(\mathbf{y}\)
Wasserstein Discriminant Analysis [11]
The function solves the following optimization problem:
\[\mathbf{P} = \mathop{\arg \min}_\mathbf{P} \quad \frac{\sum\limits_i W(P \mathbf{X}^i, P \mathbf{X}^i)}{\sum\limits_{i, j \neq i} W(P \mathbf{X}^i, P \mathbf{X}^j)}\]
where :
\(P\) is a linear projection operator in the Stiefel(p, d) manifold
\(W\) is entropic regularized Wasserstein distances
\(\mathbf{X}^i\) are samples in the dataset corresponding to class i
Choosing a Sinkhorn solver
By default and when using a regularization parameter that is not too small the default sinkhorn solver should be enough. If you need to use a small regularization to get sparse cost matrices, you should use the ot.dr.sinkhorn_log() solver that will avoid numerical errors, but can be slow in practice.
X (ndarray, shape (n, d)) – Training samples.
y (ndarray, shape (n,)) – Labels for training samples.
p (int, optional) – Size of dimensionality reduction.
reg (float, optional) – Regularization term >0 (entropic regularization)
solver (None | str, optional) – None for steepest descent or ‘TrustRegions’ for trust regions algorithm else should be a pymanopt.solvers
sinkhorn_method (str) – method used for the Sinkhorn solver, either ‘sinkhorn’ or ‘sinkhorn_log’
P0 (ndarray, shape (d, p)) – Initial starting point for projection.
normalize (bool, optional) – Normalize the Wasserstaiun distance by the average distance on P0 (default : False)
verbose (int, optional) – Print information along iterations.
P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters
proj (callable) – Projection function including mean centering.
References
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