Arccosine kernel (#245)

* WIP

* first go

* nice test

---------

Signed-off-by: Thomas Pinder <tompinder@live.co.uk>
Co-authored-by: Thomas Pinder <tompinder@live.co.uk>

gpjax/kernels/__init__.py

@@ -29,6 +29,7 @@
 )
 from gpjax.kernels.non_euclidean import GraphKernel
 from gpjax.kernels.nonstationary import (
+    ArcCosine,
     Linear,
     Polynomial,
 )
@@ -45,6 +46,7 @@
 
 __all__ = [
     "AbstractKernel",
+    "ArcCosine",
     "RBF",
     "GraphKernel",
     "Matern12",

gpjax/kernels/nonstationary/__init__.py

@@ -13,7 +13,8 @@
 # limitations under the License.
 # ==============================================================================
 
+from gpjax.kernels.nonstationary.arccosine import ArcCosine
 from gpjax.kernels.nonstationary.linear import Linear
 from gpjax.kernels.nonstationary.polynomial import Polynomial
 
-__all__ = ["Linear", "Polynomial"]
+__all__ = ["Linear", "Polynomial", "ArcCosine"]

gpjax/kernels/nonstationary/arccosine.py

@@ -0,0 +1,117 @@
+# Copyright 2022 The JaxGaussianProcesses 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,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+# ==============================================================================
+
+from dataclasses import dataclass
+
+from beartype.typing import Union
+import jax.numpy as jnp
+from jaxtyping import Float
+from simple_pytree import static_field
+import tensorflow_probability.substrates.jax.bijectors as tfb
+
+from gpjax.base import param_field
+from gpjax.kernels.base import AbstractKernel
+from gpjax.typing import (
+    Array,
+    ScalarFloat,
+    ScalarInt,
+)
+
+
+@dataclass
+class ArcCosine(AbstractKernel):
+    """The ArCosine kernel. This kernel is non-stationary and resembles the behavior
+    of neural networks. See Section 3.1 of https://arxiv.org/pdf/1112.3712.pdf for
+    additional details.
+    """
+
+    order: ScalarInt = static_field(0)
+    variance: ScalarFloat = param_field(jnp.array(1.0), bijector=tfb.Softplus())
+    weight_variance: Union[ScalarFloat, Float[Array, " D"]] = param_field(
+        jnp.array(1.0), bijector=tfb.Softplus()
+    )
+    bias_variance: ScalarFloat = param_field(jnp.array(1.0), bijector=tfb.Softplus())
+
+    def __post_init__(self):
+        if self.order not in [0, 1, 2]:
+            raise ValueError("ArcCosine kernel only implemented for orders 0, 1 and 2.")
+
+        self.name = f"ArcCosine (order {self.order})"
+
+    def __call__(self, x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
+        """Evaluate the kernel on a pair of inputs :math:`(x, y)`
+
+        Args:
+            x (Float[Array, "D"]): The left hand argument of the kernel function's
+                call.
+            y (Float[Array, "D"]): The right hand argument of the kernel function's
+                call
+
+        Returns
+        -------
+            ScalarFloat: The value of :math:`k(x, y)`.
+        """
+
+        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 *= self.variance / jnp.pi
+
+        return K.squeeze()
+
+    def _weighted_prod(
+        self, x: Float[Array, " D"], y: Float[Array, " D"]
+    ) -> ScalarFloat:
+        """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(self.weight_variance * x, y) + self.bias_variance
+
+    def _J(self, theta: ScalarFloat) -> ScalarFloat:
+        """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
+            )

tests/test_kernels/test_nonstationary.py

@@ -14,10 +14,7 @@
 # ==============================================================================
 
 from dataclasses import is_dataclass
-from itertools import (
-    permutations,
-    product,
-)
+from itertools import product
 from typing import List
 
 import jax
@@ -31,13 +28,11 @@
 from gpjax.kernels.base import AbstractKernel
 from gpjax.kernels.computations import DenseKernelComputation
 from gpjax.kernels.nonstationary import (
+    ArcCosine,
     Linear,
     Polynomial,
 )
-from gpjax.linops import (
-    LinearOperator,
-    identity,
-)
+from gpjax.linops import LinearOperator
 
 # Enable Float64 for more stable matrix inversions.
 config.update("jax_enable_x64", True)
@@ -101,7 +96,9 @@ def test_initialization(self, fields: dict, dim: int) -> None:
         # Check meta leaves
         meta = kernel._pytree__meta
         assert not any(f in meta for f in self.static_fields)
-        assert list(meta.keys()) == sorted(set(fields) - set(self.static_fields))
+        assert sorted(list(meta.keys())) == sorted(
+            set(fields) - set(self.static_fields)
+        )
 
         for field in meta:
             # Bijectors
@@ -170,3 +167,52 @@ class TestPolynomial(BaseTestKernel):
     static_fields = ["degree"]
     params = {"test_initialization": fields}
     default_compute_engine = DenseKernelComputation
+
+
+class TestArcCosine(BaseTestKernel):
+    kernel = ArcCosine
+    fields = prod(
+        {
+            "variance": [0.1, 1.0],
+            "order": [0, 1, 2],
+            "weight_variance": [0.1, 1.0],
+            "bias_variance": [0.1, 1.0],
+        }
+    )
+    static_fields = ["order"]
+    params = {"test_initialization": fields}
+    default_compute_engine = DenseKernelComputation
+
+    @pytest.mark.parametrize("order", [-1, 3], ids=lambda x: f"order={x}")
+    def test_defaults(self, order: int) -> None:
+        with pytest.raises(ValueError):
+            self.kernel(order=order)
+
+    @pytest.mark.parametrize("order", [0, 1, 2], ids=lambda x: f"order={x}")
+    def test_values_by_monte_carlo_in_special_case(self, order: int) -> None:
+        """For certain values of weight variance (1.0) and bias variance (0.0), we can test
+        our calculations using the Monte Carlo expansion of the arccosine kernel, e.g.
+        see Eq. (1) of https://cseweb.ucsd.edu/~saul/papers/nips09_kernel.pdf.
+        """
+        kernel: AbstractKernel = self.kernel(
+            weight_variance=jnp.array([1.0, 1.0]), bias_variance=1e-25, order=order
+        )
+        key = jr.PRNGKey(123)
+
+        # Inputs close(ish) together
+        a = jnp.array([[0.0, 0.0]])
+        b = jnp.array([[2.0, 2.0]])
+
+        # calc cross-covariance exactly
+        Kab_exact = kernel.cross_covariance(a, b)
+
+        # calc cross-covariance using samples
+        weights = jax.random.normal(key, (10_000, 2))  # [S, d]
+        weights_a = jnp.matmul(weights, a.T)  # [S, 1]
+        weights_b = jnp.matmul(weights, b.T)  # [S, 1]
+        H_a = jnp.heaviside(weights_a, 0.5)
+        H_b = jnp.heaviside(weights_b, 0.5)
+        integrands = H_a * H_b * (weights_a**order) * (weights_b**order)
+        Kab_approx = 2.0 * jnp.mean(integrands)
+
+        assert jnp.max(Kab_approx - Kab_exact) < 1e-4