Merge montecarlo/hutchinson and decomp/lanczos modules

Makefile

@@ -20,6 +20,7 @@ clean:
 	rm -rf *.egg-info
 	rm -rf dist site build htmlcov
 	rm -rf *.ipynb_checkpoints
+	rm matfree/_version.py
 
 doc:
 	mkdocs build

README.md

@@ -51,7 +51,7 @@ Import matfree and JAX, and set up a test problem.
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo, slq
+>>> from matfree import hutchinson, slq
 
 >>> A = jnp.reshape(jnp.arange(12.0), (6, 2))
 >>>
@@ -65,7 +65,7 @@ Estimate the trace of the matrix:
 
 ```python
 >>> key = jax.random.PRNGKey(1)
->>> normal = montecarlo.normal(shape=(2,))
+>>> normal = hutchinson.normal(shape=(2,))
 >>> trace = hutchinson.trace(matvec, key=key, sample_fun=normal)
 >>>
 >>> print(jnp.round(trace))

docs/benchmarks/control_variates.py

@@ -3,7 +3,7 @@
 Runtime: ~10 seconds.
 """
 
-from matfree import benchmark_util, hutchinson, montecarlo
+from matfree import benchmark_util, hutchinson
 from matfree.backend import func, linalg, np, plt, prng, progressbar
 
 
@@ -22,7 +22,7 @@ def f(x):
     _, jvp = func.linearize(f, x0)
     J = func.jacfwd(f)(x0)
     trace = linalg.trace(J)
-    sample_fun = montecarlo.normal(shape=(n,), dtype=float)
+    sample_fun = hutchinson.normal(shape=(n,), dtype=float)
 
     return (jvp, trace, J), (key, sample_fun)
 

docs/benchmarks/jacobian_squared.py

@@ -1,5 +1,5 @@
 """What is the fastest way of computing trace(A^5)."""
-from matfree import benchmark_util, hutchinson, montecarlo, slq
+from matfree import benchmark_util, hutchinson, slq
 from matfree.backend import func, linalg, np, plt, prng
 from matfree.backend.progressbar import progressbar
 
@@ -20,7 +20,7 @@ def f(x):
     J = func.jacfwd(f)(x0)
     A = J @ J @ J @ J
     trace = linalg.trace(A)
-    sample_fun = montecarlo.normal(shape=(n,), dtype=float)
+    sample_fun = hutchinson.normal(shape=(n,), dtype=float)
 
     def Av(v):
         return jvp(jvp(jvp(jvp(v))))

docs/control_variates.md

@@ -8,13 +8,13 @@ Imports:
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo, slq
+>>> from matfree import hutchinson, slq
 
 >>> a = jnp.reshape(jnp.arange(12.0), (6, 2))
 >>> key = jax.random.PRNGKey(1)
 
 >>> matvec = lambda x: a.T @ (a @ x)
->>> sample_fun = montecarlo.normal(shape=(2,))
+>>> sample_fun = hutchinson.normal(shape=(2,))
 
 ```
 

docs/higher_moments.md

@@ -3,13 +3,13 @@
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo, slq
+>>> from matfree import hutchinson, slq
 
 >>> a = jnp.reshape(jnp.arange(12.0), (6, 2))
 >>> key = jax.random.PRNGKey(1)
 
 >>> mvp = lambda x: a.T @ (a @ x)
->>> sample_fun = montecarlo.normal(shape=(2,))
+>>> sample_fun = hutchinson.normal(shape=(2,))
 
 ```
 
@@ -21,7 +21,7 @@ Compute them as such
 
 ```python
 >>> a = jnp.reshape(jnp.arange(36.0), (6, 6)) / 36
->>> normal = montecarlo.normal(shape=(6,))
+>>> normal = hutchinson.normal(shape=(6,))
 >>> mvp = lambda x: a.T @ (a @ x) + x
 >>> first, second = hutchinson.trace_moments(mvp, key=key, sample_fun=normal)
 >>> print(jnp.round(first, 1))
@@ -53,7 +53,7 @@ Implement this as follows:
 
 ```python
 >>> a = jnp.reshape(jnp.arange(36.0), (6, 6)) / 36
->>> sample_fun = montecarlo.normal(shape=(6,))
+>>> sample_fun = hutchinson.normal(shape=(6,))
 >>> num_samples = 10_000
 >>> mvp = lambda x: a.T @ (a @ x) + x
 >>> first, second = hutchinson.trace_moments(

docs/index.md

@@ -51,7 +51,7 @@ Import matfree and JAX, and set up a test problem.
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo, slq
+>>> from matfree import hutchinson, slq
 
 >>> A = jnp.reshape(jnp.arange(12.0), (6, 2))
 >>>
@@ -65,7 +65,7 @@ Estimate the trace of the matrix:
 
 ```python
 >>> key = jax.random.PRNGKey(1)
->>> normal = montecarlo.normal(shape=(2,))
+>>> normal = hutchinson.normal(shape=(2,))
 >>> trace = hutchinson.trace(matvec, key=key, sample_fun=normal)
 >>>
 >>> print(jnp.round(trace))

docs/log_determinants.md

@@ -6,21 +6,21 @@ Imports:
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo, slq
+>>> from matfree import hutchinson, slq
 
 >>> a = jnp.reshape(jnp.arange(12.0), (6, 2))
 >>> key = jax.random.PRNGKey(1)
 
 >>> matvec = lambda x: a.T @ (a @ x)
->>> sample_fun = montecarlo.normal(shape=(2,))
+>>> sample_fun = hutchinson.normal(shape=(2,))
 
 ```
 
 
 Estimate log-determinants as such:
 ```python
 >>> a = jnp.reshape(jnp.arange(36.0), (6, 6)) / 36
->>> sample_fun = montecarlo.normal(shape=(6,))
+>>> sample_fun = hutchinson.normal(shape=(6,))
 >>> matvec = lambda x: a.T @ (a @ x) + x
 >>> order = 3
 >>> logdet = slq.logdet_spd(order, matvec, key=key, sample_fun=sample_fun)
@@ -37,7 +37,7 @@ on arithmetic with $B$; no need to assemble $M$:
 
 ```python
 >>> a = jnp.reshape(jnp.arange(36.0), (6, 6)) / 36 + jnp.eye(6)
->>> sample_fun = montecarlo.normal(shape=(6,))
+>>> sample_fun = hutchinson.normal(shape=(6,))
 >>> matvec = lambda x: (a @ x)
 >>> vecmat = lambda x: (a.T @ x)
 >>> order = 3

docs/pytree_logdeterminants.md

@@ -11,7 +11,7 @@ Imports:
 >>> import jax.flatten_util  # this is important!
 >>> import jax.numpy as jnp
 >>>
->>> from matfree import slq, montecarlo
+>>> from matfree import slq, hutchinson
 
 ```
 Create a test-problem: a function that maps a pytree (dict) to a pytree (tuple).
@@ -70,7 +70,7 @@ Now, we can compute the log-determinant with the flattened inputs as usual:
 ```python
 >>> # Compute the log-determinant
 >>> key = jax.random.PRNGKey(seed=1)
->>> sample_fun = montecarlo.normal(shape=f0_flat.shape)
+>>> sample_fun = hutchinson.normal(shape=f0_flat.shape)
 >>> order = 3
 >>> logdet = slq.logdet_spd(order, matvec, key=key, sample_fun=sample_fun)
 

docs/vector_calculus.md

@@ -11,7 +11,7 @@ Here is how we can implement divergences and Laplacians without forming full Jac
 ```python
 >>> import jax
 >>> import jax.numpy as jnp
->>> from matfree import hutchinson, montecarlo
+>>> from matfree import hutchinson
 
 ```
 
@@ -85,7 +85,7 @@ For large-scale problems, it may be the only way of computing Laplacians reliabl
 ```python
 >>> laplacian_dense = divergence_dense(gradient)
 >>>
->>> normal = montecarlo.normal(shape=(3,))
+>>> normal = hutchinson.normal(shape=(3,))
 >>> key = jax.random.PRNGKey(1)
 >>> laplacian_matfree = divergence_matfree(gradient, key=key, sample_fun=normal)
 >>>

matfree/decomp.py

@@ -1,10 +1,166 @@
 """Matrix decomposition algorithms."""
 
-from matfree import lanczos
-from matfree.backend import containers, control_flow, linalg
+from matfree.backend import containers, control_flow, linalg, np
 from matfree.backend.typing import Array, Callable, Tuple
 
 
+class _Alg(containers.NamedTuple):
+    """Matrix decomposition algorithm."""
+
+    init: Callable
+    """Initialise the state of the algorithm. Usually, this involves pre-allocation."""
+
+    step: Callable
+    """Compute the next iteration."""
+
+    extract: Callable
+    """Extract the solution from the state of the algorithm."""
+
+    lower_upper: Tuple[int, int]
+    """Range of the for-loop used to decompose a matrix."""
+
+
+def tridiagonal_full_reortho(depth, /):
+    """Construct an implementation of **tridiagonalisation**.
+
+    Uses pre-allocation. Fully reorthogonalise vectors at every step.
+
+    This algorithm assumes a **symmetric matrix**.
+
+    Decompose a matrix into a product of orthogonal-**tridiagonal**-orthogonal matrices.
+    Use this algorithm for approximate **eigenvalue** decompositions.
+
+    """
+
+    class State(containers.NamedTuple):
+        i: int
+        basis: Array
+        tridiag: Tuple[Array, Array]
+        q: Array
+
+    def init(init_vec: Array) -> State:
+        (ncols,) = np.shape(init_vec)
+        if depth >= ncols or depth < 1:
+            raise ValueError
+
+        diag = np.zeros((depth + 1,))
+        offdiag = np.zeros((depth,))
+        basis = np.zeros((depth + 1, ncols))
+
+        return State(0, basis, (diag, offdiag), init_vec)
+
+    def apply(state: State, Av: Callable) -> State:
+        i, basis, (diag, offdiag), vec = state
+
+        # Re-orthogonalise against ALL basis elements before storing.
+        # Note: we re-orthogonalise against ALL columns of Q, not just
+        # the ones we have already computed. This increases the complexity
+        # of the whole iteration from n(n+1)/2 to n^2, but has the advantage
+        # that the whole computation has static bounds (thus we can JIT it all).
+        # Since 'Q' is padded with zeros, the numerical values are identical
+        # between both modes of computing.
+        vec, length = _normalise(vec)
+        vec, _ = _gram_schmidt_orthogonalise_set(vec, basis)
+
+        # I don't know why, but this re-normalisation is soooo crucial
+        vec, _ = _normalise(vec)
+        basis = basis.at[i, :].set(vec)
+
+        # When i==0, Q[i-1] is Q[-1] and again, we benefit from the fact
+        #  that Q is initialised with zeros.
+        vec = Av(vec)
+        basis_vectors_previous = np.asarray([basis[i], basis[i - 1]])
+        vec, (coeff, _) = _gram_schmidt_orthogonalise_set(vec, basis_vectors_previous)
+        diag = diag.at[i].set(coeff)
+        offdiag = offdiag.at[i - 1].set(length)
+
+        return State(i + 1, basis, (diag, offdiag), vec)
+
+    def extract(state: State, /):
+        _, basis, (diag, offdiag), _ = state
+        return basis, (diag, offdiag)
+
+    return _Alg(init=init, step=apply, extract=extract, lower_upper=(0, depth + 1))
+
+
+def bidiagonal_full_reortho(depth, /, matrix_shape):
+    """Construct an implementation of **bidiagonalisation**.
+
+    Uses pre-allocation. Fully reorthogonalise vectors at every step.
+
+    Works for **arbitrary matrices**. No symmetry required.
+
+    Decompose a matrix into a product of orthogonal-**bidiagonal**-orthogonal matrices.
+    Use this algorithm for approximate **singular value** decompositions.
+    """
+    nrows, ncols = matrix_shape
+    max_depth = min(nrows, ncols) - 1
+    if depth > max_depth or depth < 0:
+        msg1 = f"Depth {depth} exceeds the matrix' dimensions. "
+        msg2 = f"Expected: 0 <= depth <= min(nrows, ncols) - 1 = {max_depth} "
+        msg3 = f"for a matrix with shape {matrix_shape}."
+        raise ValueError(msg1 + msg2 + msg3)
+
+    class State(containers.NamedTuple):
+        i: int
+        Us: Array
+        Vs: Array
+        alphas: Array
+        betas: Array
+        beta: Array
+        vk: Array
+
+    def init(init_vec: Array) -> State:
+        nrows, ncols = matrix_shape
+        alphas = np.zeros((depth + 1,))
+        betas = np.zeros((depth + 1,))
+        Us = np.zeros((depth + 1, nrows))
+        Vs = np.zeros((depth + 1, ncols))
+        v0, _ = _normalise(init_vec)
+        return State(0, Us, Vs, alphas, betas, 0.0, v0)
+
+    def apply(state: State, Av: Callable, vA: Callable) -> State:
+        i, Us, Vs, alphas, betas, beta, vk = state
+        Vs = Vs.at[i].set(vk)
+        betas = betas.at[i].set(beta)
+
+        uk = Av(vk) - beta * Us[i - 1]
+        uk, alpha = _normalise(uk)
+        uk, _ = _gram_schmidt_orthogonalise_set(uk, Us)  # full reorthogonalisation
+        uk, _ = _normalise(uk)
+        Us = Us.at[i].set(uk)
+        alphas = alphas.at[i].set(alpha)
+
+        vk = vA(uk) - alpha * vk
+        vk, beta = _normalise(vk)
+        vk, _ = _gram_schmidt_orthogonalise_set(vk, Vs)  # full reorthogonalisation
+        vk, _ = _normalise(vk)
+
+        return State(i + 1, Us, Vs, alphas, betas, beta, vk)
+
+    def extract(state: State, /):
+        _, uk_all, vk_all, alphas, betas, beta, vk = state
+        return uk_all.T, (alphas, betas[1:]), vk_all, (beta, vk)
+
+    return _Alg(init=init, step=apply, extract=extract, lower_upper=(0, depth + 1))
+
+
+def _normalise(vec):
+    length = linalg.vector_norm(vec)
+    return vec / length, length
+
+
+def _gram_schmidt_orthogonalise_set(vec, vectors):  # Gram-Schmidt
+    vec, coeffs = control_flow.scan(_gram_schmidt_orthogonalise, vec, xs=vectors)
+    return vec, coeffs
+
+
+def _gram_schmidt_orthogonalise(vec1, vec2):
+    coeff = linalg.vecdot(vec1, vec2)
+    vec_ortho = vec1 - coeff * vec2
+    return vec_ortho, coeff
+
+
 def svd(
     v0: Array, depth: int, Av: Callable, vA: Callable, matrix_shape: Tuple[int, ...]
 ):
@@ -29,7 +185,7 @@ def svd(
         Shape of the matrix involved in matrix-vector and vector-matrix products.
     """
     # Factorise the matrix
-    algorithm = lanczos.bidiagonal_full_reortho(depth, matrix_shape=matrix_shape)
+    algorithm = bidiagonal_full_reortho(depth, matrix_shape=matrix_shape)
     u, (d, e), vt, _ = decompose_fori_loop(v0, Av, vA, algorithm=algorithm)
 
     # Compute SVD of factorisation
@@ -66,7 +222,7 @@ class _DecompAlg(containers.NamedTuple):
 """Decomposition algorithm type.
 
 For example, the output of
-[matfree.lanczos.tridiagonal_full_reortho(...)][matfree.lanczos.tridiagonal_full_reortho].
+[matfree.decomp.tridiagonal_full_reortho(...)][matfree.decomp.tridiagonal_full_reortho].
 """