Riemannian optimization

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Riemannian optimization is a framework for solving optimization problems with a constraint that the solution belongs to a manifold.

Let us consider the following problem. Given some TT tensor \(A\) with large tt-ranks we would like to find a tensor \(X\) (with small prescribed tt-ranks \(r\)) which is closest to \(A\) (in the sense of Frobenius norm). Mathematically it can be written as follows:

\begin{equation*} \begin{aligned} & \underset{X}{\text{minimize}} & & \frac{1}{2}\|X - A\|_F^2 \\ & \text{subject to} & & \text{tt_rank}(X) = r \end{aligned} \end{equation*}

It is known that the set of TT tensors with elementwise fixed TT ranks forms a manifold. Thus we can solve this problem using the so called Riemannian gradient descent. Given some functional \(F\) on a manifold \(\mathcal{M}\) it is defined as

\[\hat{x}_{k+1} = x_{k} - \alpha P_{T_{x_k}\mathcal{M}} \nabla F(x_k),\]

\[ \begin{align}\begin{aligned}x_{k+1} = \mathcal{R}(\hat{x}_{k+1})\\with :math:`P_{T_{x_k}\mathcal{M}}` being the projection onto the tangent space of :math:`\mathcal{M}` at the point :math:`x_k` and :math:`\mathcal{R}` being a retraction - an operation which projects points to the manifold, and :math:`\alpha` is the learning rate.\end{aligned}\end{align} \]

We can implement this in t3f using the t3f.riemannian module. As a retraction it is convenient to use the rounding method (t3f.round).

import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
tf.set_random_seed(0)
np.random.seed(0)
%matplotlib inline

try:
    import t3f
except ImportError:
    # Install T3F if it's not already installed.
    !git clone https://github.com/Bihaqo/t3f.git
    !cd t3f; pip install .
    import t3f

sess = tf.InteractiveSession()

# Initialize A randomly, with large tt-ranks
shape = 10 * [2]
init_A = t3f.random_tensor(shape, tt_rank=16)
A = t3f.get_variable('A', initializer=init_A, trainable=False)

# Create an X variable and compute the gradient of the functional. Note that it is simply X - A.

init_X = t3f.random_tensor(shape, tt_rank=2)
X = t3f.get_variable('X', initializer=init_X)

gradF = X - A

# Let us compute the projection of the gradient onto the tangent space at X

riemannian_grad = t3f.riemannian.project(gradF, X)

# Compute the update by subtracting the Riemannian gradient
# and retracting back to the manifold
alpha = 1.0

train_step = t3f.assign(X, t3f.round(X - alpha * riemannian_grad, max_tt_rank=2))

# let us also compute the value of the functional
# to see if it is decreasing
F = 0.5 * t3f.frobenius_norm_squared(X - A)

sess.run(tf.global_variables_initializer())

log = []
for i in range(100):
    F_v, _ = sess.run([F, train_step.op])
    if i % 10 == 0:
        print (F_v)
    log.append(F_v)

81.622
58.5347
56.27
56.0832
51.7328
50.7767
50.7767
50.7767
50.7767
50.7767

It is intructive to compare the obtained result with the quasioptimum delivered by the TT-round procedure.

quasi_sol = t3f.round(A, max_tt_rank=2)

val = sess.run(0.5 * t3f.frobenius_norm_squared(quasi_sol - A))
print (val)

52.4074

We see that the value is slightly bigger than the exact minimum, but TT-round is faster and cheaper to compute, so it is often used in practice.

plt.semilogy(log, label='Riemannian gradient descent')
plt.axhline(y=val, lw=1, ls='--', color='gray', label='TT-round(A)')
plt.xlabel('Iteration')
plt.ylabel('Value of the functional')
plt.legend()

<matplotlib.legend.Legend at 0x181e365f98>