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Fall back to dense eigh in sparse_cholesky for rank-deficient PSD inputs #3308

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Fall back to dense eigh in sparse_cholesky for rank-deficient PSD inputs #3308

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16 changes: 16 additions & 0 deletions cvxpy/tests/test_linalg_utils.py
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Original file line number Diff line number Diff line change
Expand Up @@ -136,6 +136,22 @@ def test_indefinite_with_zero_diagonal(self):
with self.assertRaises(ValueError, msg=lau.SparseCholeskyMessages.INDEFINITE):
lau.sparse_cholesky(A, 0.0)

def test_rank_deficient_full_diagonal(self):
# [[1,1],[1,1]] is rank-1 PSD with no zero diagonal entries; QDLDL
# alone cannot factor it, so this exercises the eigh fallback.
A = sp.csc_array(np.array([[1.0, 1.0], [1.0, 1.0]]))
sign, L, p = lau.sparse_cholesky(A)
self.assertEqual(sign, 1.0)
self.assertEqual(L.shape, (2, 1))
self.check_gram(L[p, :], A)

def test_nsd_rank_deficient_full_diagonal(self):
A = sp.csc_array(-np.array([[1.0, 1.0], [1.0, 1.0]]))
sign, L, p = lau.sparse_cholesky(A)
self.assertEqual(sign, -1.0)
self.assertEqual(L.shape, (2, 1))
self.check_gram(L[p, :], -A)

def test_nonsingular_indefinite(self):
np.random.seed(0)
n = 5
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20 changes: 16 additions & 4 deletions cvxpy/utilities/linalg.py
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Expand Up @@ -301,7 +301,19 @@ def sparse_cholesky(A, sym_tol=settings.CHOL_SYM_TOL, assume_psd=False):

except ValueError:
raise
except Exception as e:
raise ValueError(
f"{SparseCholeskyMessages.FACTORIZATION_FAILED}: {e}"
) from e
except Exception:
# QDLDL has no pivoting and fails when a zero pivot appears
# mid-elimination on rank-deficient PSD/NSD inputs (e.g. [[1,1],[1,1]]).
# Fall back to a dense symmetric eigendecomposition.
w, V = np.linalg.eigh(A.toarray())
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@claude claude Bot Apr 29, 2026

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The except Exception clause (unchanged from before) now triggers an O(n³) dense eigendecomposition + O(n²) memory allocation via A.toarray(), rather than just re-raising as a ValueError. If a large sparse matrix hits this path for an unexpected reason (not rank-deficiency), the dense fallback could OOM or hang.

Consider either:

  1. Narrowing the catch to the specific exception type QDLDL raises on zero-pivot (likely a C-level RuntimeError), or
  2. Adding a size guard before densifying, e.g.:
if n > 5000:
    raise ValueError(f"{SparseCholeskyMessages.FACTORIZATION_FAILED}: matrix too large for dense fallback")

This way truly unexpected failures on large matrices still surface immediately rather than silently attempting dense computation.

scale = np.max(np.abs(w))
if scale == 0:
return 1.0, sp.csr_array((n, 0)), np.arange(n)
is_psd = np.all(w >= -tol * scale)
is_nsd = np.all(w <= tol * scale)
if not (is_psd or is_nsd):
raise ValueError(SparseCholeskyMessages.INDEFINITE)
sign = 1.0 if is_psd else -1.0
mask = np.abs(w) > tol * scale
L = V[:, mask] * np.sqrt(np.abs(w[mask]))
return sign, sp.csr_array(L), np.arange(n)
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