Add conditional mean and cov (#39)

* Add conditional mean and cov

* Change log_prob test to asymmetric A

* Doc

tests/test_conditional.py

@@ -0,0 +1,32 @@
+import jax
+from jax import numpy as jnp
+
+import thermox
+
+
+def test_mean_and_cov():
+    jax.config.update("jax_enable_x64", True)
+    dim = 2
+    t = 1.0
+
+    A = jnp.array([[3, 2.5], [2, 4.0]])
+    b = jax.random.normal(jax.random.PRNGKey(1), (dim,))
+    x0 = jax.random.normal(jax.random.PRNGKey(2), (dim,))
+    D = 2 * jnp.eye(dim)
+
+    mean = thermox.conditional.mean(t, x0, A, b, D)
+    samples = jax.vmap(
+        lambda k: thermox.sample(k, jnp.array([0.0, t]), x0, A, b, D)[-1]
+    )(jax.random.split(jax.random.PRNGKey(0), 1000000))
+    assert mean.shape == (dim,)
+    assert jnp.allclose(mean, jnp.mean(samples, axis=0), atol=1e-2)
+
+    cov = thermox.conditional.covariance(t, A, D)
+    assert cov.shape == (dim, dim)
+    assert jnp.allclose(cov, jnp.cov(samples.T), atol=1e-3)
+
+    mean_and_cov = thermox.conditional.mean_and_covariance(t, x0, A, b, D)
+    assert mean_and_cov[0].shape == (dim,)
+    assert mean_and_cov[1].shape == (dim, dim)
+    assert jnp.allclose(mean_and_cov[0], mean, atol=1e-5)
+    assert jnp.allclose(mean_and_cov[1], cov, atol=1e-5)

tests/test_log_prob.py

@@ -91,42 +91,40 @@ def test_MLE():
     D_true = jnp.array([[1, 0.3, -0.1], [0.3, 1, 0.2], [-0.1, 0.2, 1.0]])
 
     nts = 300
-    ts = jnp.linspace(0, 10, nts)
+    ts = jnp.linspace(0, 100, nts)
     x0 = jnp.zeros_like(b_true)
 
-    n_trajecs = 3
+    n_trajecs = 5
     rks = jax.random.split(jax.random.PRNGKey(0), n_trajecs)
 
     samps = jax.vmap(lambda key: thermox.sample(key, ts, x0, A_true, b_true, D_true))(
         rks
     )
 
-    A_sqrt_init = jnp.tril(jnp.eye(3) + jax.random.normal(rks[0], (3, 3)) * 1e-1)
+    A_init = jnp.eye(3) + jax.random.normal(rks[0], (3, 3)) * 1e-1
     b_init = jnp.zeros(3)
     D_sqrt_init = jnp.eye(3)
 
     log_prob_true = thermox.log_prob(ts, samps[0], A_true, b_true, D_true)
     log_prob_init = thermox.log_prob(
-        ts, samps[0], A_sqrt_init @ A_sqrt_init.T, b_init, D_sqrt_init @ D_sqrt_init.T
+        ts, samps[0], A_init, b_init, D_sqrt_init @ D_sqrt_init.T
     )
 
     assert log_prob_true > log_prob_init
 
     # Gradient descent
     def loss(params):
-        A_sqrt, b, D_sqrt = params
-        A_sqrt = jnp.tril(A_sqrt)
+        A, b, D_sqrt = params
         D_sqrt = jnp.tril(D_sqrt)
-        A = A_sqrt @ A_sqrt.T
         D = D_sqrt @ D_sqrt.T
         return -jax.vmap(lambda s: thermox.log_prob(ts, s, A, b, D))(
             samps
         ).mean() / len(ts)
 
     val_and_g = jax.jit(jax.value_and_grad(loss))
 
-    ps = (A_sqrt_init, b_init, D_sqrt_init)
-    ps_true = (jnp.linalg.cholesky(A_true), b_true, jnp.linalg.cholesky(D_true))
+    ps = (A_init, b_init, D_sqrt_init)
+    ps_true = (A_true, b_true, jnp.linalg.cholesky(D_true))
 
     v, g = val_and_g(ps)
     v_true, g_true = val_and_g(ps_true)
@@ -138,7 +136,7 @@ def loss(params):
     n_steps = 20000
     neg_log_probs = jnp.zeros(n_steps)
 
-    optimizer = optax.adam(1e-2)
+    optimizer = optax.adam(1e-3)
     opt_state = optimizer.init(ps)
 
     for i in range(n_steps):
@@ -149,7 +147,7 @@ def loss(params):
         ps = optax.apply_updates(ps, updates)
         neg_log_probs = neg_log_probs.at[i].set(neg_log_prob)
 
-    A_recover = ps[0] @ ps[0].T
+    A_recover = ps[0]
     b_recover = ps[1]
     D_recover = ps[2] @ ps[2].T
 

thermox/__init__.py

@@ -1,4 +1,5 @@
 from thermox import linalg
+from thermox import conditional
 from thermox.sampler import sample
 from thermox.prob import log_prob
 from thermox.utils import preprocess

thermox/conditional.py

@@ -0,0 +1,98 @@
+from jax import numpy as jnp
+from jax import Array
+
+from thermox.utils import (
+    ProcessedDriftMatrix,
+    ProcessedDiffusionMatrix,
+    handle_matrix_inputs,
+)
+from thermox.sampler import expm_vp
+
+
+def mean(
+    t: float,
+    x0: Array,
+    A: Array | ProcessedDriftMatrix,
+    b: Array,
+    D: Array | ProcessedDiffusionMatrix,
+) -> Array:
+    """Computes the mean of p(x_t | x_0)
+
+    For x_t evolving according to the SDE:
+
+    dx = - A * (x - b) dt + sqrt(D) dW
+
+    Args:
+        ts: Times at which samples are collected. Includes time for x0.
+        x0: Initial state of the process.
+        A: Drift matrix (Array or thermox.ProcessedDriftMatrix).
+            Note: If a thermox.ProcessedDriftMatrix instance is used as input,
+            must be the transformed drift matrix, A_y, given by thermox.preprocess,
+            not thermox.utils.preprocess_drift_matrix.
+        b: Drift displacement vector.
+        D: Diffusion matrix (Array or thermox.ProcessedDiffusionMatrix).
+
+    """
+    A_y, D = handle_matrix_inputs(A, D)
+
+    y0 = D.sqrt_inv @ (x0 - b)
+    return b + D.sqrt @ expm_vp(A_y, y0, t)
+
+
+def covariance(
+    t: float,
+    A: Array | ProcessedDriftMatrix,
+    D: Array | ProcessedDiffusionMatrix,
+) -> Array:
+    """Computes the covariance of p(x_t | x_0)
+
+    For x evolving according to the SDE:
+
+    dx = - A * (x - b) dt + sqrt(D) dW
+
+    Args:
+        ts: Times at which samples are collected. Includes time for x0.
+        A: Drift matrix (Array or thermox.ProcessedDriftMatrix).
+            Note: If a thermox.ProcessedDriftMatrix instance is used as input,
+            must be the transformed drift matrix, A_y, given by thermox.preprocess,
+            not thermox.utils.preprocess_drift_matrix.
+        D: Diffusion matrix (Array or thermox.ProcessedDiffusionMatrix).
+    """
+    A_y, D = handle_matrix_inputs(A, D)
+
+    identity_diffusion_cov = (
+        A_y.sym_eigvecs
+        @ jnp.diag((1 - jnp.exp(-2 * A_y.sym_eigvals * t)) / (2 * A_y.sym_eigvals))
+        @ A_y.sym_eigvecs.T
+    )
+    return D.sqrt @ identity_diffusion_cov @ D.sqrt.T
+
+
+def mean_and_covariance(
+    t: float,
+    x0: Array,
+    A: Array | ProcessedDriftMatrix,
+    b: Array,
+    D: Array | ProcessedDiffusionMatrix,
+) -> tuple[Array, Array]:
+    """Computes the mean and covariance of p(x_t | x_0)
+
+    For x evolving according to the SDE:
+
+    dx = - A * (x - b) dt + sqrt(D) dW
+
+    Args:
+        ts: Times at which samples are collected. Includes time for x0.
+        x0: Initial state of the process.
+        A: Drift matrix (Array or thermox.ProcessedDriftMatrix).
+            Note: If a thermox.ProcessedDriftMatrix instance is used as input,
+            must be the transformed drift matrix, A_y, given by thermox.preprocess,
+            not thermox.utils.preprocess_drift_matrix.
+        b: Drift displacement vector.
+        D: Diffusion matrix (Array or thermox.ProcessedDiffusionMatrix).
+
+    """
+    A, D = handle_matrix_inputs(A, D)
+    mean_val = mean(t, x0, A, b, D)
+    covariance_val = covariance(t, A, D)
+    return mean_val, covariance_val

thermox/prob.py

@@ -36,7 +36,7 @@ def log_prob(
 
     Args:
         ts: Times at which samples are collected. Includes time for x0.
-        xs: Initial state of the process.
+        xs: States of the process.
         A: Drift matrix (Array or thermox.ProcessedDriftMatrix).
             Note: If a thermox.ProcessedDriftMatrix instance is used as input,
             must be the transformed drift matrix, A_y, given by thermox.preprocess,