Source code for alibi_detect.utils.tensorflow.distance

import logging
import numpy as np
import tensorflow as tf
from typing import Callable, Tuple, List, Optional, Union

logger = logging.getLogger(__name__)


[docs]def squared_pairwise_distance(x: tf.Tensor, y: tf.Tensor, a_min: float = 1e-30, a_max: float = 1e30) -> tf.Tensor: """ TensorFlow pairwise squared Euclidean distance between samples x and y. Parameters ---------- x Batch of instances of shape [Nx, features]. y Batch of instances of shape [Ny, features]. a_min Lower bound to clip distance values. a_max Upper bound to clip distance values. Returns ------- Pairwise squared Euclidean distance [Nx, Ny]. """ x2 = tf.reduce_sum(x ** 2, axis=-1, keepdims=True) y2 = tf.reduce_sum(y ** 2, axis=-1, keepdims=True) dist = x2 + tf.transpose(y2, (1, 0)) - 2. * x @ tf.transpose(y, (1, 0)) return tf.clip_by_value(dist, a_min, a_max)
[docs]def mmd2_from_kernel_matrix(kernel_mat: tf.Tensor, m: int, permute: bool = False, zero_diag: bool = True) -> tf.Tensor: """ Compute maximum mean discrepancy (MMD^2) between 2 samples x and y from the full kernel matrix between the samples. Parameters ---------- kernel_mat Kernel matrix between samples x and y. m Number of instances in y. permute Whether to permute the row indices. Used for permutation tests. zero_diag Whether to zero out the diagonal of the kernel matrix. Returns ------- MMD^2 between the samples from the kernel matrix. """ n = kernel_mat.shape[0] - m if zero_diag: kernel_mat = kernel_mat - tf.linalg.diag(tf.linalg.diag_part(kernel_mat)) if permute: idx = np.random.permutation(kernel_mat.shape[0]) kernel_mat = tf.gather(tf.gather(kernel_mat, indices=idx, axis=0), indices=idx, axis=1) k_xx, k_yy, k_xy = kernel_mat[:-m, :-m], kernel_mat[-m:, -m:], kernel_mat[-m:, :-m] c_xx, c_yy = 1 / (n * (n - 1)), 1 / (m * (m - 1)) mmd2 = c_xx * tf.reduce_sum(k_xx) + c_yy * tf.reduce_sum(k_yy) - 2. * tf.reduce_mean(k_xy) return mmd2
[docs]def mmd2(x: tf.Tensor, y: tf.Tensor, kernel: Callable) -> float: """ Compute MMD^2 between 2 samples. Parameters ---------- x Batch of instances of shape [Nx, features]. y Batch of instances of shape [Ny, features]. kernel Kernel function. Returns ------- MMD^2 between the samples x and y. """ n, m = x.shape[0], y.shape[0] c_xx, c_yy = 1 / (n * (n - 1)), 1 / (m * (m - 1)) k_xx, k_yy, k_xy = kernel(x, x), kernel(y, y), kernel(x, y) # type: ignore return (c_xx * (tf.reduce_sum(k_xx) - tf.linalg.trace(k_xx)) + c_yy * (tf.reduce_sum(k_yy) - tf.linalg.trace(k_yy)) - 2. * tf.reduce_mean(k_xy))
[docs]def relative_euclidean_distance(x: tf.Tensor, y: tf.Tensor, eps: float = 1e-12, axis: int = -1) -> tf.Tensor: """ Relative Euclidean distance. Parameters ---------- x Tensor used in distance computation. y Tensor used in distance computation. eps Epsilon added to denominator for numerical stability. axis Axis used to compute distance. Returns ------- Tensor with relative Euclidean distance across specified axis. """ denom = tf.concat([tf.reshape(tf.norm(x, ord=2, axis=axis), (-1, 1)), tf.reshape(tf.norm(y, ord=2, axis=axis), (-1, 1))], axis=1) dist = tf.norm(x - y, ord=2, axis=axis) / (tf.reduce_min(denom, axis=axis) + eps) return dist
[docs]def permed_lsdds( k_all_c: tf.Tensor, x_perms: List[tf.Tensor], y_perms: List[tf.Tensor], H: tf.Tensor, H_lam_inv: Optional[tf.Tensor] = None, lam_rd_max: float = 0.2, return_unpermed: bool = False, ) -> Union[Tuple[tf.Tensor, tf.Tensor], Tuple[tf.Tensor, tf.Tensor, tf.Tensor]]: """ Compute LSDD estimates from kernel matrix across various ref and test window samples Parameters ---------- k_all_c Kernel matrix of simmilarities between all samples and the kernel centers. x_perms List of B reference window index vectors y_perms List of B test window index vectors H Special (scaled) kernel matrix of simmilarities between kernel centers H_lam_inv Function of H corresponding to a particular regulariation parameter lambda. See Eqn 11 of Bu et al. (2017) lam_rd_max The maximum relative difference between two estimates of LSDD that the regularization parameter lambda is allowed to cause. Defaults to 0.2. Only relavent if H_lam_inv is not supplied. return_unpermed Whether or not to return value corresponding to unpermed order defined by k_all_c Returns ------- Vector of B LSDD estimates for each permutation, H_lam_inv which may have been inferred, and optionally the unpermed LSDD estimate. """ # Compute (for each bootstrap) the average distance to each kernel center (Eqn 7) k_xc_perms = tf.stack([tf.gather(k_all_c, x_inds) for x_inds in x_perms], axis=0) k_yc_perms = tf.stack([tf.gather(k_all_c, y_inds) for y_inds in y_perms], axis=0) h_perms = tf.reduce_mean(k_xc_perms, axis=1) - tf.reduce_mean(k_yc_perms, axis=1) if H_lam_inv is None: # We perform the initialisation for multiple candidate lambda values and pick the largest # one for which the relative difference (RD) between two difference estimates is below lambda_rd_max. # See Appendix A candidate_lambdas = [1/(4**i) for i in range(10)] # TODO: More principled selection H_plus_lams = tf.stack([H+tf.eye(H.shape[0], dtype=H.dtype)*can_lam for can_lam in candidate_lambdas], axis=0) H_plus_lam_invs = tf.transpose(tf.linalg.inv(H_plus_lams), [1, 2, 0]) # lambdas last omegas = tf.einsum('jkl,bk->bjl', H_plus_lam_invs, h_perms) # (Eqn 8) h_omegas = tf.einsum('bj,bjl->bl', h_perms, omegas) omega_H_omegas = tf.einsum('bkl,bkl->bl', tf.einsum('bjl,jk->bkl', omegas, H), omegas) rds = tf.reduce_mean(1 - (omega_H_omegas/h_omegas), axis=0) less_than_rd_inds = tf.where(rds < lam_rd_max) if len(less_than_rd_inds) == 0: repeats = k_all_c.shape[0] - np.unique(k_all_c, dim=0).shape[0] if repeats > 0: msg = "Too many repeat instances for LSDD-based detection. \ Try using MMD-based detection instead" else: msg = "Unknown error. Try using MMD-based detection instead" raise ValueError(msg) lambda_index = int(less_than_rd_inds[0]) lam = candidate_lambdas[lambda_index] logger.info(f"Using lambda value of {lam:.2g} with RD of {float(rds[lambda_index]):.2g}") H_plus_lam_inv = tf.linalg.inv(H+lam*tf.eye(H.shape[0], dtype=H.dtype)) H_lam_inv = 2*H_plus_lam_inv - (tf.transpose(H_plus_lam_inv, [1, 0]) @ H @ H_plus_lam_inv) # (blw Eqn 11) # Now to compute an LSDD estimate for each permutation lsdd_perms = tf.reduce_sum( h_perms * tf.transpose(H_lam_inv @ tf.transpose(h_perms, [1, 0]), [1, 0]), axis=1 ) # (Eqn 11) if return_unpermed: n_x = x_perms[0].shape[0] h = tf.reduce_mean(k_all_c[:n_x], axis=0) - tf.reduce_mean(k_all_c[n_x:], axis=0) lsdd_unpermed = tf.reduce_sum(h[None, :] * tf.transpose(H_lam_inv @ h[:, None], [1, 0])) return lsdd_perms, H_lam_inv, lsdd_unpermed else: return lsdd_perms, H_lam_inv