# Copyright 2022 The thomaspinder Contributors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
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# See the License for the specific language governing permissions and
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# ==============================================================================
import beartype.typing as tp
import equinox as eqx
import jax.numpy as jnp
from jaxtyping import Float
from paramax import AbstractUnwrappable
from gpjax.kernels.base import AbstractKernel, _val
from gpjax.kernels.computations import (
AbstractKernelComputation,
DenseKernelComputation,
)
from gpjax.parameters import (
NonNegativeReal,
)
from gpjax.typing import (
Array,
ScalarArray,
ScalarFloat,
)
WeightVariance = tp.Union[Float[Array, "D"], ScalarArray]
WeightVarianceCompatible = tp.Union[ScalarFloat, list[float], WeightVariance]
[docs]
class ArcCosine(AbstractKernel):
r"""The ArCosine kernel.
This kernel is non-stationary and resembles the behavior of neural networks.
See Section 3.1 of
[Cho and Saul (2011)](https://arxiv.org/abs/1112.3712) for
additional details.
"""
order: tp.Literal[0, 1, 2] = eqx.field(static=True, default=0)
variance: AbstractUnwrappable
weight_variance: AbstractUnwrappable
bias_variance: AbstractUnwrappable
name = "ArcCosine"
def __init__(
self,
active_dims: tp.Union[list[int], slice, None] = None,
order: tp.Literal[0, 1, 2] = 0,
variance: tp.Union[ScalarFloat, AbstractUnwrappable] = 1.0,
weight_variance: tp.Union[WeightVarianceCompatible, AbstractUnwrappable] = 1.0,
bias_variance: tp.Union[ScalarFloat, AbstractUnwrappable] = 1.0,
n_dims: tp.Union[int, None] = None,
compute_engine: AbstractKernelComputation = DenseKernelComputation(),
):
"""Initializes the kernel.
Args:
active_dims: The indices of the input dimensions that the kernel operates on.
order: The order of the kernel. Must be 0, 1 or 2.
variance: The variance of the kernel σ.
weight_variance: The weight variance of the kernel.
bias_variance: The bias variance of the kernel.
n_dims: The number of input dimensions. If `lengthscale` is an array, this
argument is ignored.
compute_engine: The computation engine that the kernel uses to compute the
covariance matrix.
"""
if order not in [0, 1, 2]:
raise ValueError("ArcCosine kernel only implemented for orders 0, 1 and 2.")
self.order = order
def _as_nonneg(value):
if isinstance(value, AbstractUnwrappable):
return value
return NonNegativeReal(value)
self.weight_variance = _as_nonneg(weight_variance)
self.bias_variance = _as_nonneg(bias_variance)
self.variance = _as_nonneg(variance)
super().__init__(active_dims, n_dims, compute_engine)
def __call__(self, x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarArray:
x = self.slice_input(x)
y = self.slice_input(y)
x_x = self._weighted_prod(x, x)
x_y = self._weighted_prod(x, y)
y_y = self._weighted_prod(y, y)
cos_theta = x_y / jnp.sqrt(x_x * y_y)
jitter = 1e-15 # improve numerical stability
theta = jnp.arccos(jitter + (1 - 2 * jitter) * cos_theta)
K = self._J(theta)
K *= jnp.sqrt(x_x) ** self.order
K *= jnp.sqrt(y_y) ** self.order
K *= _val(self.variance) / jnp.pi
return K.squeeze()
def _weighted_prod(
self, x: Float[Array, " D"], y: Float[Array, " D"]
) -> ScalarFloat:
r"""Calculate the weighted product between two arguments.
Args:
x (Float[Array, "D"]): The left hand argument.
y (Float[Array, "D"]): The right hand argument.
Returns:
ScalarFloat: The value of the weighted product between the two arguments``.
"""
return jnp.inner(_val(self.weight_variance) * x, y) + _val(self.bias_variance)
def _J(self, theta: ScalarFloat) -> ScalarFloat:
r"""Evaluate the angular dependency function corresponding to the desired order.
Args:
theta (Float[Array, "1"]): The weighted angle between inputs.
Returns:
Float[Array, "1"]: The value of the angular dependency function`.
"""
if self.order == 0:
return jnp.pi - theta
elif self.order == 1:
return jnp.sin(theta) + (jnp.pi - theta) * jnp.cos(theta)
else:
return 3.0 * jnp.sin(theta) * jnp.cos(theta) + (jnp.pi - theta) * (
1.0 + 2.0 * jnp.cos(theta) ** 2
)