Notebooks · lib/randla/trace

Trace estimation from matrix-vector products by Hutchinson and Hutch++

Both estimators see the matrix only through apply(X), which returns A X for an n-by-s block X: a def or a lambda, so A can be an operator never stored (a Hessian, a matrix function applied by Lanczos, an inverse applied by a solver).

hutchinson_fn(apply, n, m, seed, stream): (1/m) sum of w^T A w over m Rademacher vectors w. Unbiased, with variance 2 (|A|_F^2 - sum a_ii^2) / m, so the error falls like 1/sqrt(m): 100 times the products for one more digit.
hutchpp_fn(apply, n, m, seed, stream): Hutch++ (Meyer, Musco, Musco and Woodruff, SOSA 2021). A third of the budget finds the dominant range Q of A (from A S), the trace of Q^T A Q is taken exactly, and Hutchinson runs on the rest, (I - Q Q^T) A (I - Q Q^T), whose Frobenius norm is small when A's spectrum decays. For PSD A the error falls like 1/m.

When not to use them: for a matrix you hold, trace(A) is exact and costs n reads. When the spectrum is flat (A near a multiple of I), Hutch++ spends a third of its products on a range that captures nothing, and plain Hutchinson with all m products does better.

Used with from randla.trace import ....

Look inside this program

Atlas
The program’s map: every function and which calls which.
There’s no recorded run of this program yet.

Shift+Enter in a cell runs it and moves to the next. Running a cell runs every cell above it too, as one program.

[1]lines 28–92

from randla.sketch import gaussian, rademacher
from randla.dense import orth

#| Hutchinson's estimate of tr(A) from m Rademacher probes, A given by
#| apply(X) = A X.
def hutchinson_fn(apply, n: Int, m: Int, seed: Int, stream: Int) -> Num {
    W = rademacher(n, m, seed, stream)
    return sum(W * apply(W)) / float(m)
}

#| Hutch++'s estimate of tr(A) from m products with A (m / 3 for the range,
#| m / 3 for Q^T A Q, the rest for Hutchinson on the deflated matrix).
def hutchpp_fn(apply, n: Int, m: Int, seed: Int, stream: Int) -> Num {
    #: a third of the queries find the dominant range Q
    s: Int = m // 3
    g: Int = m - 2 * s
    S = rademacher(n, s, seed, stream)
    Q = orth(apply(S))
    #: the trace on Q, exactly
    t1 = sum(Q * apply(Q))
    #: Hutchinson on the rest, its queries projected off Q
    G = rademacher(n, g, seed, stream + 1)
    G = G - Q @ (transpose(Q) @ G)
    t2 = sum(G * apply(G)) / float(g)
    return t1 + t2
}

#| Hutchinson's estimate for a matrix A.
def hutchinson(A: Arr, m: Int, seed: Int, stream: Int) -> Num {
    n: Int = int(shape(A)[0])
    return hutchinson_fn(fn(X) -> A @ X, n, m, seed, stream)
}

#| Hutch++'s estimate for a matrix A.
def hutchpp(A: Arr, m: Int, seed: Int, stream: Int) -> Num {
    n: Int = int(shape(A)[0])
    return hutchpp_fn(fn(X) -> A @ X, n, m, seed, stream)
}

test "Hutch++ beats Hutchinson on a decaying spectrum, by more as m grows" {
    n: Int = 150
    V = orth(gaussian(n, n, 31, 0))
    lam = pow(arange(1.0, float(n) + 1.0), -2.0)
    A = (V * lam) @ transpose(V)
    t = sum(lam)
    trials: Int = 6
    e1_small = 0.0
    e2_small = 0.0
    e1_big = 0.0
    e2_big = 0.0
    for r to trials {
        e1_small += abs(hutchinson(A, 30, r, 0) - t) / t
        e2_small += abs(hutchpp(A, 30, r, 0) - t) / t
        e1_big += abs(hutchinson(A, 240, r, 0) - t) / t
        e2_big += abs(hutchpp(A, 240, r, 0) - t) / t
    }
    show e1_small / trials, e2_small / trials, e1_big / trials, e2_big / trials
    assert_lt(e2_small, e1_small)
    assert_lt(e2_big, 0.1 * e1_big)
    # eight times the products: Hutchinson gains about sqrt(8), Hutch++ more
    assert_lt(e2_big, e2_small / 6.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

from randla.sketch import gaussian, rademacher
from randla.dense import orth

#| Hutchinson's estimate of tr(A) from m Rademacher probes, A given by
#| apply(X) = A X.
def hutchinson_fn(apply, n: Int, m: Int, seed: Int, stream: Int) -> Num {
    W = rademacher(n, m, seed, stream)
    return sum(W * apply(W)) / float(m)
}

#| Hutch++'s estimate of tr(A) from m products with A (m / 3 for the range,
#| m / 3 for Q^T A Q, the rest for Hutchinson on the deflated matrix).
def hutchpp_fn(apply, n: Int, m: Int, seed: Int, stream: Int) -> Num {
    #: a third of the queries find the dominant range Q
    s: Int = m // 3
    g: Int = m - 2 * s
    S = rademacher(n, s, seed, stream)
    Q = orth(apply(S))
    #: the trace on Q, exactly
    t1 = sum(Q * apply(Q))
    #: Hutchinson on the rest, its queries projected off Q
    G = rademacher(n, g, seed, stream + 1)
    G = G - Q @ (transpose(Q) @ G)
    t2 = sum(G * apply(G)) / float(g)
    return t1 + t2
}

#| Hutchinson's estimate for a matrix A.
def hutchinson(A: Arr, m: Int, seed: Int, stream: Int) -> Num {
    n: Int = int(shape(A)[0])
    return hutchinson_fn(fn(X) -> A @ X, n, m, seed, stream)
}

#| Hutch++'s estimate for a matrix A.
def hutchpp(A: Arr, m: Int, seed: Int, stream: Int) -> Num {
    n: Int = int(shape(A)[0])
    return hutchpp_fn(fn(X) -> A @ X, n, m, seed, stream)
}

test "Hutch++ beats Hutchinson on a decaying spectrum, by more as m grows" {
    n: Int = 150
    V = orth(gaussian(n, n, 31, 0))
    lam = pow(arange(1.0, float(n) + 1.0), -2.0)
    A = (V * lam) @ transpose(V)
    t = sum(lam)
    trials: Int = 6
    e1_small = 0.0
    e2_small = 0.0
    e1_big = 0.0
    e2_big = 0.0
    for r to trials {
        e1_small += abs(hutchinson(A, 30, r, 0) - t) / t
        e2_small += abs(hutchpp(A, 30, r, 0) - t) / t
        e1_big += abs(hutchinson(A, 240, r, 0) - t) / t
        e2_big += abs(hutchpp(A, 240, r, 0) - t) / t
    }
    show e1_small / trials, e2_small / trials, e1_big / trials, e2_big / trials
    assert_lt(e2_small, e1_small)
    assert_lt(e2_big, 0.1 * e1_big)
    # eight times the products: Hutchinson gains about sqrt(8), Hutch++ more
    assert_lt(e2_big, e2_small / 6.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors