Make SLQ integrands behave correctly for arbitrary input vectors (#174)

* Make integrand_slq_spd behave correctly for arbitrary input vectors, not just unit-second-moment input vectors

* Make integrand_slq_product behave correctly, too

matfree/slq.py

@@ -25,7 +25,8 @@ def integrand_slq_spd(matfun, order, matvec, /):
 
     def quadform(v0, *parameters):
         v0_flat, v_unflatten = tree_util.ravel_pytree(v0)
-        v0_flat /= linalg.vector_norm(v0_flat)
+        length = linalg.vector_norm(v0_flat)
+        v0_flat /= length
 
         def matvec_flat(v_flat):
             v = v_unflatten(v_flat)
@@ -50,10 +51,8 @@ def matvec_flat(v_flat):
 
         # Since Q orthogonal (orthonormal) to v0, Q v = Q[0],
         # and therefore (Q v)^T f(D) (Qv) = Q[0] * f(diag) * Q[0]
-        (dim,) = v0_flat.shape
-
         fx_eigvals = func.vmap(matfun)(eigvals)
-        return dim * linalg.vecdot(eigvecs[0, :], fx_eigvals * eigvecs[0, :])
+        return length**2 * linalg.vecdot(eigvecs[0, :], fx_eigvals * eigvecs[0, :])
 
     return quadform
 
@@ -86,7 +85,8 @@ def integrand_slq_product(matfun, depth, matvec, vecmat, /):
 
     def quadform(v0, *parameters):
         v0_flat, v_unflatten = tree_util.ravel_pytree(v0)
-        v0_flat /= linalg.vector_norm(v0_flat)
+        length = linalg.vector_norm(v0_flat)
+        v0_flat /= length
 
         def matvec_flat(v_flat):
             v = v_unflatten(v_flat)
@@ -115,10 +115,9 @@ def vecmat_flat(w_flat):
 
         # Since Q orthogonal (orthonormal) to v0, Q v = Q[0],
         # and therefore (Q v)^T f(D) (Qv) = Q[0] * f(diag) * Q[0]
-        _, ncols = matrix_shape
         eigvals, eigvecs = S**2, Vt.T
         fx_eigvals = func.vmap(matfun)(eigvals)
-        return ncols * linalg.vecdot(eigvecs[0, :], fx_eigvals * eigvecs[0, :])
+        return length**2 * linalg.vecdot(eigvecs[0, :], fx_eigvals * eigvecs[0, :])
 
     return quadform
 

tests/test_slq/test_logdet_product.py

@@ -38,3 +38,33 @@ def vecmat(x):
     expected = linalg.slogdet(A.T @ A)[1]
     print_if_assert_fails = ("error", np.abs(received - expected), "target:", expected)
     assert np.allclose(received, expected, atol=1e-2, rtol=1e-2), print_if_assert_fails
+
+
+@testing.parametrize("n", [50])
+# usually: ~1.5 * num_significant_eigvals.
+# But logdet seems to converge sooo much faster.
+def test_logdet_product_exact_for_full_order_lanczos(n):
+    r"""Computing v^\top f(A^\top @ A) v with max-order Lanczos is exact for _any_ v."""
+    # Construct a (numerically nice) matrix
+    singular_values = np.sqrt(np.arange(1.0, 1.0 + n, step=1.0))
+    A = test_util.asymmetric_matrix_from_singular_values(
+        singular_values, nrows=n, ncols=n
+    )
+
+    # Set up max-order Lanczos approximation inside SLQ for the matrix-logarithm
+    order = n - 1
+    integrand = slq.integrand_logdet_product(order, lambda v: A @ v, lambda v: v @ A)
+
+    # Construct a vector without that does not have expected 2-norm equal to "dim"
+    x = prng.normal(prng.prng_key(seed=1), shape=(n,)) + 1
+
+    # Compute v^\top @ log(A) @ v via Lanczos
+    received = integrand(x)
+
+    # Compute the "true" value of v^\top @ log(A) @ v via eigenvalues
+    eigvals, eigvecs = linalg.eigh(A.T @ A)
+    logA = eigvecs @ linalg.diagonal_matrix(np.log(eigvals)) @ eigvecs.T
+    expected = x.T @ logA @ x
+
+    # They should be identical
+    assert np.allclose(received, expected)

tests/test_slq/test_logdet_spd.py

@@ -36,3 +36,31 @@ def matvec(x):
     expected = linalg.slogdet(A)[1]
     print_if_assert_fails = ("error", np.abs(received - expected), "target:", expected)
     assert np.allclose(received, expected, atol=1e-2, rtol=1e-2), print_if_assert_fails
+
+
+@testing.parametrize("n", [50])
+# usually: ~1.5 * num_significant_eigvals.
+# But logdet seems to converge sooo much faster.
+def test_logdet_spd_exact_for_full_order_lanczos(n):
+    r"""Computing v^\top f(A) v with max-order Lanczos should be exact for _any_ v."""
+    # Construct a (numerically nice) matrix
+    eigvals = np.arange(1.0, 1.0 + n, step=1.0)
+    A = test_util.symmetric_matrix_from_eigenvalues(eigvals)
+
+    # Set up max-order Lanczos approximation inside SLQ for the matrix-logarithm
+    order = n - 1
+    integrand = slq.integrand_logdet_spd(order, lambda v: A @ v)
+
+    # Construct a vector without that does not have expected 2-norm equal to "dim"
+    x = prng.normal(prng.prng_key(seed=1), shape=(n,)) + 10
+
+    # Compute v^\top @ log(A) @ v via Lanczos
+    received = integrand(x)
+
+    # Compute the "true" value of v^\top @ log(A) @ v via eigenvalues
+    eigvals, eigvecs = linalg.eigh(A)
+    logA = eigvecs @ linalg.diagonal_matrix(np.log(eigvals)) @ eigvecs.T
+    expected = x.T @ logA @ x
+
+    # They should be identical
+    assert np.allclose(received, expected)