# ot.dr

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

## Functions

ot.dr.dist(x1, x2)[source]

Compute squared euclidean distance between samples (autograd)

ot.dr.ewca(X, U0=None, reg=1, k=2, method='BCD', sinkhorn_method='sinkhorn', stopThr=1e-06, maxiter=100, maxiter_sink=1000, maxiter_MM=10, verbose=0)[source]

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

Parameters:
• 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.

Returns:

• pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters.

• U (ndarray, shape (d, k)) – Matrix Stiefel manifold.

References

### Examples using ot.dr.ewca

Entropic Wasserstein Component Analysis

Entropic Wasserstein Component Analysis
ot.dr.fda(X, y, p=2, reg=1e-16)[source]

Fisher Discriminant Analysis

Parameters:
• 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 (ridge regularization)

Returns:

• P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters

• proj (callable) – projection function including mean centering

### Examples using ot.dr.fda

Wasserstein Discriminant Analysis

Wasserstein Discriminant Analysis
ot.dr.logsumexp(M, axis)[source]

Log-sum-exp reduction compatible with autograd (no numpy implementation)

ot.dr.projection_robust_wasserstein(X, Y, a, b, tau, U0=None, reg=0.1, k=2, stopThr=0.001, maxiter=100, verbose=0, random_state=None)[source]

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

Parameters:
• 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.

Returns:

• pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters

• U (ndarray, shape (d, k)) – Projection operator.

References

ot.dr.sinkhorn(w1, w2, M, reg, k)[source]

Sinkhorn algorithm with fixed number of iteration (autograd)

ot.dr.sinkhorn_log(w1, w2, M, reg, k)[source]

Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)

ot.dr.split_classes(X, y)[source]

split samples in $$\mathbf{X}$$ by classes in $$\mathbf{y}$$

ot.dr.wda(X, y, p=2, reg=1, k=10, solver=None, sinkhorn_method='sinkhorn', maxiter=100, verbose=0, P0=None, normalize=False)[source]

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.

Parameters:
• 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.

Returns:

• P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters

• proj (callable) – Projection function including mean centering.

References

### Examples using ot.dr.wda

Wasserstein Discriminant Analysis

Wasserstein Discriminant Analysis
ot.dr.dist(x1, x2)[source]

Compute squared euclidean distance between samples (autograd)

ot.dr.ewca(X, U0=None, reg=1, k=2, method='BCD', sinkhorn_method='sinkhorn', stopThr=1e-06, maxiter=100, maxiter_sink=1000, maxiter_MM=10, verbose=0)[source]

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

Parameters:
• 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.

Returns:

• pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters.

• U (ndarray, shape (d, k)) – Matrix Stiefel manifold.

References

ot.dr.fda(X, y, p=2, reg=1e-16)[source]

Fisher Discriminant Analysis

Parameters:
• 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 (ridge regularization)

Returns:

• P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters

• proj (callable) – projection function including mean centering

ot.dr.logsumexp(M, axis)[source]

Log-sum-exp reduction compatible with autograd (no numpy implementation)

ot.dr.projection_robust_wasserstein(X, Y, a, b, tau, U0=None, reg=0.1, k=2, stopThr=0.001, maxiter=100, verbose=0, random_state=None)[source]

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

Parameters:
• 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.

Returns:

• pi (ndarray, shape (n, n)) – Optimal transportation matrix for the given parameters

• U (ndarray, shape (d, k)) – Projection operator.

References

ot.dr.sinkhorn(w1, w2, M, reg, k)[source]

Sinkhorn algorithm with fixed number of iteration (autograd)

ot.dr.sinkhorn_log(w1, w2, M, reg, k)[source]

Sinkhorn algorithm in log-domain with fixed number of iteration (autograd)

ot.dr.split_classes(X, y)[source]

split samples in $$\mathbf{X}$$ by classes in $$\mathbf{y}$$

ot.dr.wda(X, y, p=2, reg=1, k=10, solver=None, sinkhorn_method='sinkhorn', maxiter=100, verbose=0, P0=None, normalize=False)[source]

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.

Parameters:
• 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.

Returns:

• P (ndarray, shape (d, p)) – Optimal transportation matrix for the given parameters

• proj (callable) – Projection function including mean centering.

References