[ADD] Vanilla Hutchinson trace estimation (#38)

* [ADD] Hutchinson trace estimator

* [DOC] Add Hutchinson estimator to documentation

* [REF] Move example, polish rst

* [RTD] Try fixing deprecated `build.image`

https://blog.readthedocs.com/use-build-os-config/#use-build-os-instead-of-build-image-on-your-configuration-file

* [FIX] Precision of doctest

.readthedocs.yaml

@@ -6,10 +6,14 @@ version: 2
 sphinx:
   configuration: docs/rtd/conf.py
 
+build:
+  os: ubuntu-22.04
+  tools:
+    python: "3.8"
+
 python:
-  version: "3.8"
   install:
-    - method: pip
-      path: .
-      extra_requirements:
-        - docs
+  - method: pip
+    path: .
+    extra_requirements:
+      - docs

curvlinops/__init__.py

@@ -13,6 +13,7 @@
     lanczos_approximate_spectrum,
 )
 from curvlinops.submatrix import SubmatrixLinearOperator
+from curvlinops.trace.hutchinson import HutchinsonTraceEstimator
 
 __all__ = [
     "HessianLinearOperator",
@@ -28,4 +29,5 @@
     "lanczos_approximate_log_spectrum",
     "LanczosApproximateSpectrumCached",
     "LanczosApproximateLogSpectrumCached",
+    "HutchinsonTraceEstimator",
 ]

curvlinops/trace/__init__.py

@@ -0,0 +1 @@
+"""Trace estimation techniques."""

curvlinops/trace/hutchinson.py

@@ -0,0 +1,85 @@
+"""Vanilla Hutchinson trace estimation."""
+
+from typing import Callable, Dict
+
+from numpy import dot, ndarray
+from scipy.sparse.linalg import LinearOperator
+
+from curvlinops.trace.sampling import normal, rademacher
+
+
+class HutchinsonTraceEstimator:
+    """Class to perform trace estimation with Hutchinson's method.
+
+    For details, see
+
+    - Hutchinson, M. (1989). A stochastic estimator of the trace of the influence
+      matrix for laplacian smoothing splines. Communication in Statistics---Simulation
+      and Computation.
+
+    Example:
+        >>> from numpy import trace, mean, round
+        >>> from numpy.random import rand, seed
+        >>> seed(0) # make deterministic
+        >>> A = rand(10, 10)
+        >>> tr_A = trace(A) # exact trace as reference
+        >>> estimator = HutchinsonTraceEstimator(A)
+        >>> # one- and multi-sample approximations
+        >>> tr_A_low_precision = estimator.sample()
+        >>> tr_A_high_precision = mean([estimator.sample() for _ in range(1_000)])
+        >>> assert abs(tr_A - tr_A_low_precision) > abs(tr_A - tr_A_high_precision)
+        >>> round(tr_A, 4), round(tr_A_low_precision, 4), round(tr_A_high_precision, 4)
+        (4.4575, 6.6796, 4.3886)
+
+    Attributes:
+        SUPPORTED_DISTRIBUTIONS: Dictionary mapping supported distributions to their
+            sampling functions.
+    """
+
+    SUPPORTED_DISTRIBUTIONS: Dict[str, Callable[[int], ndarray]] = {
+        "rademacher": rademacher,
+        "normal": normal,
+    }
+
+    def __init__(self, A: LinearOperator):
+        """Store the linear operator whose trace will be estimated.
+
+        Args:
+            A: Linear square-shaped operator whose trace will be estimated.
+
+        Raises:
+            ValueError: If the operator is not square.
+        """
+        if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
+            raise ValueError(f"A must be square. Got shape {A.shape}.")
+        self._A = A
+
+    def sample(self, distribution: str = "rademacher") -> float:
+        """Draw a sample from the trace estimator.
+
+        Multiple samples can be combined into a more accurate trace estimation via
+        averaging.
+
+        Args:
+            distribution: Distribution of the vector along which the linear operator
+                will be evaluated. Either ``'rademacher'`` or ``'normal'``.
+                Default is ``'rademacher'``.
+
+        Returns:
+            Sample from the trace estimator.
+
+        Raises:
+            ValueError: If the distribution is not supported.
+        """
+        dim = self._A.shape[1]
+
+        if distribution not in self.SUPPORTED_DISTRIBUTIONS:
+            raise ValueError(
+                f"Unsupported distribution {distribution:!r}. "
+                f"Supported distributions are {list(self.SUPPORTED_DISTRIBUTIONS)}."
+            )
+
+        v = self.SUPPORTED_DISTRIBUTIONS[distribution](dim)
+        Av = self._A @ v
+
+        return dot(v, Av)

curvlinops/trace/sampling.py

@@ -0,0 +1,29 @@
+"""Sampling methods for random vectors."""
+
+from numpy import ndarray
+from numpy.random import binomial, randn
+
+
+def rademacher(dim: int) -> ndarray:
+    """Draw a vector with i.i.d. Rademacher elements.
+
+    Args:
+        dim: Dimension of the vector.
+
+    Returns:
+        Vector with i.i.d. Rademacher elements and specified dimension.
+    """
+    num_trials, success_prob = 1, 0.5
+    return binomial(num_trials, success_prob, size=dim).astype(float) * 2 - 1
+
+
+def normal(dim: int) -> ndarray:
+    """Drawa vector with i.i.d. standard normal elements.
+
+    Args:
+        dim: Dimension of the vector.
+
+    Returns:
+        Vector with i.i.d. standard normal elements and specified dimension.
+    """
+    return randn(dim)

docs/rtd/linops.rst

@@ -59,3 +59,9 @@ Spectral density approximation
 
 .. autoclass:: curvlinops.LanczosApproximateSpectrumCached
    :members: __init__, approximate_spectrum
+
+Trace approximation
+===================
+
+.. autoclass:: curvlinops.HutchinsonTraceEstimator
+   :members: __init__, sample

makefile

@@ -69,10 +69,10 @@ install-test:
 .PHONY: test test-light
 
 test:
-	@pytest -vx --run-optional-tests=montecarlo --cov=curvlinops test
+	@pytest -vx --run-optional-tests=montecarlo --cov=curvlinops --doctest-modules curvlinops test
 
 test-light:
-	@pytest -vx --cov=curvlinops test
+	@pytest -vx --cov=curvlinops --doctest-modules curvlinops test
 
 .PHONY: install-lint
 

test/trace/__init__.py

@@ -0,0 +1 @@
+"""Test ``curvlinops.trace``."""

test/trace/test_hutchinson.py

@@ -0,0 +1,43 @@
+"""Test ``curvlinops.trace.__init__``."""
+
+from numpy import isclose, mean, trace
+from numpy.random import rand, seed
+from pytest import mark
+
+from curvlinops import HutchinsonTraceEstimator
+
+DISTRIBUTIONS = ["rademacher", "normal"]
+DISTRIBUTION_IDS = [f"distribution={distribution}" for distribution in DISTRIBUTIONS]
+
+
+@mark.parametrize("distribution", DISTRIBUTIONS, ids=DISTRIBUTION_IDS)
+def test_HutchinsonTraceEstimator(distribution: str):
+    """Test whether Hutchinon's trace estimator converges to the true trace.
+
+    Args:
+        distribution: Distribution of the random vectors used for the trace estimation.
+    """
+    seed(0)
+    A = rand(10, 10)
+    tr_A = trace(A)
+
+    samples = []
+    max_samples = 20_000
+    chunk_size = 2_000  # add that many new samples before comparing against the truth
+    atol, rtol = 1e-3, 1e-2
+
+    estimator = HutchinsonTraceEstimator(A)
+
+    while len(samples) < max_samples:
+        samples.extend(
+            [estimator.sample(distribution=distribution) for _ in range(chunk_size)]
+        )
+        tr_estimator = mean(samples)
+        if not isclose(tr_A, tr_estimator, atol=atol, rtol=rtol):
+            print(f"{len(samples)} samples: Tr(A)={tr_A:.5f}≠{tr_estimator:.5f}.")
+        else:
+            # quit once the estimator has converged
+            break
+
+    tr_estimator = mean(samples)
+    assert isclose(tr_A, tr_estimator, atol=atol, rtol=rtol)