cholgraph

JAX & PyTensor -native sparse Cholesky via CHOLMOD. Solves with symmetric positive definite sparse matrices run at full native speed inside @jax.jit (and lax.scan / lax.fori_loop) — no Python callback overhead.

Why?

No open-source JIT framework (JAX, PyTorch, TensorFlow) exposes sparse Cholesky as a compilable primitive; klujax covers sparse LU, which is ~2× slower than Cholesky for SPD systems.
The target workload is Gibbs samplers (e.g. Bayesian spatial econometrics), where an SPD precision matrix is solved thousands of times with the same sparsity pattern but changing values, inside a JIT-compiled loop.
Benchmarks (M-series macOS, 2D grid Laplacian, values changing every iteration): matches a hand-written scikit-sparse Python loop per iteration and is ~2.7× faster than scipy.sparse.linalg.splu, while running entirely inside jax.jit.

How it works

solve and logdet are XLA FFI custom calls into CHOLMOD. The extension caches symbolic analyses (fill-reducing ordering + elimination tree) keyed on the sparsity pattern, so repeated calls with the same pattern only pay for the numeric refactorization — and calls with unchanged values skip even that, sharing one factorization between solve and logdet. There are no handles to manage and nothing to pass through JIT boundaries; the caching is transparent.

Installation

conda env create -f environment.yml   # suitesparse, jax, nanobind, cmake, ...
conda activate cholgraph
pip install --no-build-isolation .

Quick start

import jax
import jax.numpy as jnp
import numpy as np
import cholgraph

jax.config.update("jax_enable_x64", True)   # required: CHOLMOD is float64

# SPD matrix in COO form. Entries with Ai <= Aj are used (upper triangle);
# pass the full symmetric matrix or just its upper triangle.
Ai = np.array([0, 0, 1, 1, 1, 2, 2], dtype=np.int32)
Aj = np.array([0, 1, 0, 1, 2, 1, 2], dtype=np.int32)
Ax = jnp.array([4.0, 1.0, 1.0, 5.0, 2.0, 2.0, 6.0])
b = jnp.array([1.0, 2.0, 3.0])

x = cholgraph.solve(Ai, Aj, Ax, b)          # eager
ld = cholgraph.logdet(Ai, Aj, Ax, n=3)      # log|A| from the same factorization

@jax.jit                                      # ...or fully JIT-compiled
def gibbs_step(Ax, b):
    x = cholgraph.solve(Ai, Aj, Ax, b)      # full CHOLMOD speed, no callbacks
    ld = cholgraph.logdet(Ai, Aj, Ax, n=3)  # factorization shared with solve
    return x, ld

Features:

jit / lax.scan: the symbolic analysis is computed once and reused across iterations.
Autodiff: solve has a custom VJP (reverse-mode) in both Ax and b; logdet has one in Ax (via the selected inverse — see below). Together they give the gradient of a Gaussian log-density, so the precision matrix's values can be fit by gradient-based inference (HMC/NUTS, empirical Bayes).
vmap: jax.vmap(solve) lowers to a single native FFI call that loops over the batch in C++ (reusing the cached analysis), rather than XLA per-iteration dispatch. Composes with grad (vmap(grad(solve)) batches too). Map over Ax, b, or both.
Multiple right-hand sides: b may be (n,) or (n, n_rhs).
Factor-part solves: mode=cholgraph.MODE_LT etc. expose CHOLMOD's solve systems (P' L L' P = A). Sampling y ~ N(0, A^{-1}): y = solve(..., solve(..., z, mode=MODE_LT), mode=MODE_PT).
Not positive definite → runtime exception (the factor is always a true LL').

Factor once, do everything: `factor_solve` / `sample_gaussian`

solve and logdet are separate primitives, so a Gibbs sweep that needs a posterior mean, a correlated draw, and a log-determinant factors the same A several times — and under vmap that is one factorization per solve per batch element. factor_solve factors A once and serves every requested solve (each a chain of MODE_* codes) plus an optional logdet from that single factor. Under vmap it lowers to one batched FFI call that factors once per element, whatever the number of chains.

# Gibbs Gaussian step: posterior mean, a draw ~ N(mean, A^-1), and log|A|,
# from ONE factorization. eta = mean + P' L^-T z  (since A = P' L L' P).
eta, mean, ld = cholgraph.sample_gaussian(Ai, Aj, Ax, b, z, want_logdet=True)

# ...or spell it out with the general primitive:
(mean, w), ld = cholgraph.factor_solve(
    Ai, Aj, Ax,
    [(b, cholgraph.MODE_A),                          # A^-1 b
     (z, (cholgraph.MODE_LT, cholgraph.MODE_PT))],  # P' L^-T z  (chain)
    want_logdet=True)
eta = mean + w

Each rhs entry is (b, modes) where modes is one MODE_* or a sequence applied left to right. cholgraph.factorization_count() reports how many real factorizations have happened — handy for confirming the fusion. Benchmarked Gibbs draw (mean + sample + logdet) vmapped over a batch of different A's: 4× fewer factorizations and ~3.3–3.6× faster than issuing the separate solve/logdet primitives. factor_solve is forward-only (no autodiff rule); use solve/logdet when you need gradients.

Gradients & the selected inverse

solve and logdet are the two halves of a Gaussian log-density's gradient, so a precision matrix A(θ) with a fixed pattern and θ-dependent values can be fit by gradient-based inference — HMC/NUTS (e.g. via numpyro/blackjax, or PyMC's JAX sampling backend), VI, or empirical-Bayes/MAP optimization:

def neg_log_post(Ax):                                  # up to constants
    quad = b @ cholgraph.solve(Ai, Aj, Ax, b)         # b' A^-1 b   (solve VJP)
    return 0.5 * quad - 0.5 * cholgraph.logdet(Ai, Aj, Ax, n)   # log|A| (logdet VJP)

grad_Ax = jax.grad(neg_log_post)(Ax)                   # works under jit / vmap

logdet's reverse-mode rule uses that d log|A| / dA = A^{-1}, evaluated only at A's sparsity pattern by Takahashi's selected-inversion recurrence over the Cholesky factor — never the dense inverse. That quantity is exposed directly:

z = cholgraph.selinv(Ai, Aj, Ax, n)   # z[k] == (A^-1)[Ai[k], Aj[k]]
var = z[Ai == Aj]                      # diag(A^-1): Gaussian marginal variances

selinv shares the factorization cache, is JIT-compilable and vmap-able, and costs one selected-inversion pass over the factor (O(nnz(L))-ish), not n solves. factor_solve / sample_gaussian remain forward-only.

PyMC / PyTensor (NUTS)

The same CHOLMOD core is exposed as a PyTensor frontend for PyMC's default backend, so gradient-based samplers (NUTS) can differentiate through the sparse solve and log-determinant without going through JAX/XLA. It's an optional extra — the base package stays JAX-only:

pip install "cholgraph[pytensor]"

import pytensor.tensor as pt
import cholgraph.pytensor as cjpt

Ax = pt.dvector("Ax")                 # the precision-matrix values (θ-dependent)
# Gaussian log-density (up to constants); grad flows into Ax
logp = -0.5 * pt.dot(b, cjpt.solve(Ai, Aj, Ax, b)) + 0.5 * cjpt.logdet(Ai, Aj, Ax, n)
g = pt.grad(logp, Ax)                 # solve VJP + logdet (selected-inverse) VJP

cjpt.solve, cjpt.logdet, and cjpt.selinv mirror the JAX functions and carry the same reverse-mode rules (solve in Ax/b, logdet in Ax); the pattern (Ai, Aj) is non-differentiable data. On PyTensor ≥ 3.1 the gradient routes through Op.pullback; older versions use grad.

Use the C backend, not numba

These Ops implement a pure-Python perform (it calls the native core, which holds no GIL and does the real work — the CHOLMOD call dominates). PyTensor's C backend (FAST_RUN, the default) calls that perform directly with no penalty. PyTensor's numba backend cannot JIT a Python perform, so it falls back to object mode and prints a UserWarning on every call — functionally correct but with per-call overhead. Prefer the C backend:

import pytensor

pytensor.config.mode = "FAST_RUN"     # C backend (default); avoids numba object mode
# For PyMC, this is the default; if you sample through the numba/JAX linker
# instead, drive NUTS via the JAX frontend (numpyro/blackjax) rather than these Ops.

Concretely: keep PyMC on its default sampler (C backend) to use these Ops, and switch to the JAX frontend above if you deliberately run PyMC's JAX/numba linker.

JAX sparse (`BCOO`)

JAX's native sparse type is jax.experimental.sparse.BCOO, whose .indices is (nnz, 2) and .data is (nnz,). Convenience wrappers accept one directly:

from jax.experimental import sparse as jsparse
A = jsparse.BCOO.fromdense(A_dense)         # or build however you like

x  = cholgraph.solve_bcoo(A, b)            # == solve(A.indices[:,0], A.indices[:,1], A.data, b)
ld = cholgraph.logdet_bcoo(A)
x  = cholgraph.update_solve_bcoo(A, c, b)  # rank-k update, as below

The analysis-reuse speedup is unaffected: the pattern cache keys on the concrete index values (exactly a BCOO's .indices), so a stable pattern across jit/vmap calls keeps hitting the cache. A full-symmetric BCOO works directly (only the upper triangle is read), and unsorted/duplicate entries are handled. Only a plain 2D BCOO (n_batch=0, n_dense=0) is supported.

Rank-k update / downdate

update_solve solves (A ± C C') x = b by applying CHOLMOD's cholmod_updown to a working copy of A's cached factor, instead of refactoring the modified matrix from scratch. A is factored once; each call is O(k · path) where path is the elimination-tree path touched by C's nonzeros.

# Add an observation (rank-1, sparse update column) and re-solve, cheaply:
x = cholgraph.update_solve(Ai, Aj, Ax, c, b)                 # (A + c c') x = b
x = cholgraph.update_solve(Ai, Aj, Ax, c, b, downdate=True)  # (A - c c') x = b
x, ld = cholgraph.update_solve(Ai, Aj, Ax, C, b, return_logdet=True)  # C is (n, k)

When it pays off: the update column(s) C must be sparse (a few nonzeros — e.g. one data point and its neighbors). On the grid-Laplacian benchmark a rank-1 sparse update is ~3× faster than a full factorize+solve. A dense C walks the whole tree and is slower than refactoring — use plain solve on the reassembled matrix in that case. The base cached factor is never mutated, so update_solve is a pure function (works under jit; not differentiable).

Options

cholgraph.set_options(supernodal="simplicial")  # or "auto" (default), "supernodal"
cholgraph.clear_cache()                         # free cached factorizations

For very sparse matrices (e.g. planar/grid graphs), "simplicial" often gives faster triangular solves; "supernodal" (BLAS-based) wins on denser problems. "auto" lets CHOLMOD choose based on the matrix.

Status / roadmap

solve (all CHOLMOD solve modes), logdet, symbolic + numeric caching, custom VJP, multi-RHS, tests, benchmarks
Native batching: jax.vmap(solve) → one FFI call looping over the batch in C++
cholmod_updown rank-k update/downdate (update_solve)
Cache the simplicial LDL' base factor for updown (rebuilt only on refactor), so the LL'→LDL' conversion is paid once per base change, not once per call
factor_solve / sample_gaussian: factor once, serve many solve chains + logdet from one factor; fuses under vmap to one factorization per batch element
Differentiable logdet (reverse-mode in Ax) and the selinv selected inverse (Takahashi recurrence over the factor); pairs with solve's VJP for full Gaussian log-density gradients
PyTensor frontend (cholgraph.pytensor, optional extra) with matching autodiff, for PyMC's default backend / NUTS — a second frontend over the same CHOLMOD core
float32 (CHOLMOD 5 single precision) and int64 indices
Autodiff rule for factor_solve (currently forward-only)
Wheels / conda-forge packaging

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This release

0.6.0 This release

Jul 20, 2026 13 files

cholgraph 0.6.0

cholgraph

Why?

How it works

Installation

Quick start

Factor once, do everything: factor_solve / sample_gaussian

Gradients & the selected inverse

PyMC / PyTensor (NUTS)

Use the C backend, not numba

JAX sparse (BCOO)

Rank-k update / downdate

Options

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1 maintainer

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Factor once, do everything: `factor_solve` / `sample_gaussian`

JAX sparse (`BCOO`)