Stochastic Estimation with $Z_2$ Noise

K.F. Liu; S.J. Dong

REVIEW

Stochastic Estimation with $Z_2$ Noise

S.J. Dong, K.F. Liu

High Energy Physics - Lattice · Submitted 25 Aug 1993 · 12 references

Pith's verdict: The Z2 probe is the durable contribution and it holds up; the quoted 1.54x/2.4x superiority factor does not follow from the printed inputs.

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arXiv IDhep-lat/9308015 (version 1)

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CitationK.F. Liu, S.J. Dong. "Stochastic Estimation with $Z_2$ Noise." arXiv:hep-lat/9308015 (1993). https://pith.science/paper/JZ6JIKOE

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Editor's note

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Short version: the Z2 probe is the real contribution and it holds up. The quantitative '2.4x fewer configurations' number does not follow from what's on the page, and that's the one thing I'd push back on in review. Everything else is in good shape for a 1993 letter of this length. What's new: they take the existing stochastic estimator (pseudofermion and Gaussian noise, refs [1] and [2]) and swap in a plus/minus one microcanonical noise. The argument is that the noise-dependent piece of the variance goes as C2 squared, i.e. - 1, which vanishes only for a two-point distribution. That is Cauchy-Schwarz and it is correct. Table 1 backs it directly: sigma_2 falls 1.49(1) to 0.43(1) to 0.28(1) to 0.00(0) across Gaussian, the two double-hump distributions, and Z2, while sigma_1 stays flat near 0.78 for all four. This is the Rademacher probe people now use without citing anyone, and it's worth knowing where it came from. Credit where due: they are honest about scope. They say plainly that the method dies for far off-diagonal quantities where the signal falls exponentially and the noise does not, and they do not claim general quark propagators. The near-diagonal trace regime is stated as the domain of validity rather than oversold. Soft spots, in order of weight. First, Eq. (5) is quoted from ref [9], a King College preprint listed as KC/93-01 and unpublished at the time, so the bridge from C2 = 0 to minimum variance is cited rather than derived. It's a two-line argument and should have been shown, but it's presentational, not a hole. Second, and this is the one that matters: Eq. (5) is per matrix element, while the trace estimator is a single random variable, not a sum of independent per-element estimators. The paper says 'extending eq. (5) to the variance of the trace' and never displays the extension. Summing naively over the diagonal with their own 0.8 ratio gives about 1.87, not 1.54; plausible trace formulas bracket 1.54 but don't reproduce it. The qualitative claim is untouched, but the headline number is under-supported. Third, the 0.8 ratio comes from one column of one gauge configuration, extended to all columns by translational invariance and never checked. Minor on its own, but it is the input to the previous point. Fourth, the Z2 sigma_2 = 0.00(0) row is an identity, not an independent measurement; don't read it as confirmation. The determinant, eigenvalue and eigenvector section is a one-paragraph outlook with no results, and should be treated as such. Who this is for: anyone doing stochastic trace or log-determinant estimation, lattice or otherwise. The Z2 result is the part that travels. Recommendation: send it out. The core claim is right and this is the origin point for a tool people now use routinely. Ask for the trace-variance derivation in revision.

Referee report

Significance

The central observation is correct and, properly stated, is a small theorem rather than a numerical preference: for any unit-variance white noise, <eta^4> >= <eta^2>^2 = 1 by Cauchy-Schwarz, with equality if and only if eta^2 = 1 almost surely, so C2 = 0 is the provable minimum and Z2 noise is optimal in that class. The paper deserves credit for identifying this cleanly, for testing it against three competing noises in Table 1 rather than asserting it, and for benchmarking against an exact conjugate-gradient inversion of a column of the quark matrix, which is external to the stochastic estimator. The error-scaling predictions (1/sqrt(L) in the number of noise vectors, a further 1/sqrt(N) for the trace from translational, colour and spin symmetry) are falsifiable and are checked against the numerical data at the stated level. The authors are also candid about the limitation of the method: they state explicitly that for |x-y| >> a the signal falls exponentially while the error stays constant, so the method is not useful for general quark propagators. That acknowledged limitation is appropriate and does not need to be argued further; what remains unresolved is the quantitative claim,

Major objections (3)

[Text following Eq. (5), and the 1.54 / 2.4 factor] The quantitative superiority factor is under-supported as written. Eq. (5) is stated for a single matrix element M^{-1}_ij, but the text then says 'extending eq. (5) to the variance of the trace' without displaying the extension, and quotes a standard-deviation ratio of 1.54 between Gaussian and Z2 noise (hence 2.4 times more Gaussian configurations). The trace estimator T = sum_i eta_i X_i is a single random variable, not a sum of independent per-element estimators, so its variance is not simply sum_i Var[M^{-1}_ii]; the correct expression involves the full Frobenius-type sum plus cross terms. If one instead sums Eq. (5) over diagonal elements using the stated ratio sum_{k != 1}[M^{-1}_k1]^2/[M^{-1}_11]^2 = 0.8 and the Gaussian value C2^2 = 2, one obtains (2 + 0.8)/0.8 = 3.5, i.e. a factor 1.87, not 1.54. The quoted number therefore either relies on unstated assumptions about the trace variance or contains an arithmetic slip. Since this factor is the paper's one quantitative statement of practical advantage, the trace variance formula should be written out explicitly and the arithmetic shown, or the number corrected.
[Eq. (5) and its attribution to Ref. [9]] The variance decomposition of Eq. (5) is the load-bearing result of the paper, and it is attributed to Ref. [9], which is listed as an unpublished King College preprint (KC/93-01). The result is short enough to derive in two lines by expanding E[(eta_j X_i - M^{-1}_ij)^2] with the white-noise conditions of Eq. (1), and doing so would make the paper self-contained and would also make transparent exactly which assumptions enter the trace extension discussed above. As it stands, a reader cannot check the central formula without obtaining an unpublished preprint.
[Text preceding Table 1 and the paragraph quoting the 0.8 ratio] The numerical input to the accuracy estimate, sum_{k != 1}[M^{-1}_k1]^2/[M^{-1}_11]^2 = 0.8, is obtained from inverting a single column of a single gauge configuration, and is then extended to all columns 'based on translational invariance'. No error estimate is given for this ratio, and no second column, second configuration, or second kappa is checked. Because the 1.54 factor depends directly on this number, and because the ratio is a ratio of sums over the whole lattice (so translational invariance constrains it only on average, not configuration by configuration), at least one independent cross-check should be reported, or the quoted factor should be presented with an explicit caveat that it is an estimate from one configuration.

Minor objections (7)

[Table 1 versus the preceding text] The text describes the double-hump distributions as eta^2 exp(-eta^6/2) and eta^4 exp(-eta^10/2), while Table 1 lists eta^2 exp(-eta^4/2) and eta^4 exp(-eta^10/2). The two should agree; please correct whichever is wrong.
[Table 1, Z2 row] The entry sigma_2 = 0.00(0) for Z2 noise is an identity, not an independent measurement, since C2 vanishes exactly for a ±1 distribution. It would be clearer to say so in the caption rather than presenting it alongside the fitted values for the other noises.
[Paragraph following Eq. (5) (error-to-signal estimate)] The quoted 'error to signal ratio to be sqrt(1.5 x 10^-3)' is ambiguous: it is unclear whether the square root applies to 1.5 or to 1.5 x 10^-3, and the two readings differ by more than an order of magnitude. Given the stated N = 1.18 x 10^6 and ratio 0.8, the variance ratio is about 1.5 x 10^-3 and the rms error-to-signal ratio is about 4%. Please state which quantity is meant.
[Broken cross-reference in the paragraph after Eq. (5)] The text contains the literal string 'eq. (refeq.variance)', an unresolved LaTeX reference. This should be replaced by the correct equation number.
[Fig. 1 and the 5838 saving factor] Fig. 1 is described as showing accumulated averages for 'a gauge configuration', i.e. a single configuration, and the histograms are over noise vectors only, so they do not represent gauge-to-gauge fluctuations; the caption should say this explicitly. Similarly, the factor 5838 should be stated as N/L under the assumptions that each noise solve costs the same as one column inversion and that all N columns would otherwise be required.
[Applications paragraph (Refs. [10], [11])] The statements that preliminary results on the pi-N sigma term and the flavour-singlet axial charge g_A^0 are 'encouraging' rest on Refs. [10] and [11], the latter listed as 'under preparation'. Since these are the paper's main physics applications, a sentence summarizing what was actually computed and with what statistical error would strengthen the case considerably.
[Conclusion] The word 'optimal' should be qualified once, in the conclusion as well as in the body: Z2 is optimal within the class of uncorrelated unit-variance white noises, in the sense that C2 = 0 is the Cauchy-Schwarz minimum. The paper mentions Z_N noise in passing; a single sentence stating the class and the one-line argument would make the optimality claim precise rather than rhetorical.

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The paper

The paper claims

The paper argues that a random noise made only of plus and minus one is the best probe for estimating entries and traces of a large matrix inverse, because one of the two error terms in the stochastic estimate vanishes exactly for that噪声.

The reading

Solving $M X = �eta$ for many random vectors $�eta$ and averaging $�eta_j X_i$ gives an unbiased estimate of the inverse matrix element $M^{-1}_{ij}$; the question is which distribution of $�eta$ makes the estimate least noisy for a fixed number of vectors. The paper proposes the $Z_2$ noise, whose entries are independently $+1$ or $-1$ with equal probability, and argues that it is optimal among white noises because its diagonal error $C_2=�sqrt{�langle �eta^4�rangle-1}$ is exactly zero. A variance formula decomposes the error into a term proportional to $C_2^2$ and a term that is the same for every white noise, so the $Z_2$ choice removes the only noise-dependent part. In a quenched lattice QCD example the authors estimate that reaching the same accuracy with Gaussian noise would require about $2.4$ times as many noise vectors, and they use the method for quark loops, the chiral condensate, and topological susceptibility while noting that it fails for far off-diagonal propagators.

What carries the argument

The load-bearing object is the $Z_2$ noise: each component of each noise vector is independently chosen as $+1$ or $-1$ with equal probability, so $�langle�eta_i�rangle=0$, $�langle�eta_i�eta_j�rangle=�delta_{ij}$, and $�langle�eta_i^4�rangle=1$. That last identity makes the diagonal error $C_2=�sqrt{�langle�eta^4�rangle-1}$ vanish, which in turn kills the $[M^{-1}_{ij}]^2 C_2^2$ term in the variance formula. The paper's efficiency argument therefore rests on a moment identity plus a variance decomposition, not on a special property of QCD; the QCD calculation supplies the demonstration and the size of the practical gain.

Extended reading notes

Core claim

The central claim is that the $Z_2$ microcanonical noise—random $�pm1$ entries—is the optimal noise for stochastic matrix inversion among distributions satisfying $�langle �eta_i�rangle=0$ and $�langle �eta_i�eta_j�rangle=�delta_{ij}$. The authors measure two errors, the average off-diagonal overlap $C_1$ and the diagonal deviation $C_2$, and find at $N=500$ that $C_1$ is essentially the same for Gaussian, double-hump, and $Z_2$ noises, while $C_2$ falls from $1.49(1)$ for Gaussian to $0.00(0)$ for $Z_2$. Using the variance decomposition $�rm Var}[M^{-1}_{ij}]=�rac{1}{L}�{[M^{-1}_{ij}]^2 C_2^2+�sum_{k�ne j}[M^{-1}_{ik}]^2�}$, they conclude that $Z_2$ gives the minimum variance because its $C_2$ term vanishes, and they apply this to diagonal and near-diagonal traces of the quark matrix in lattice QCD.

Load-bearing premise

The quantitative gain—about $2.4$ times fewer Gaussian noise vectors for the same accuracy—rests on a single measured ratio $�sum_{k�ne1}[M^{-1}_{k1}]^2/[M^{-1}_{11}]^2=0.8$ from one column of one gauge configuration, which the authors extend to all columns by translational invariance without checking another column, configuration, or quark mass.

Editorial extensions

If this is right

For any matrix and any white noise, the $Z_2$ choice removes the diagonal-error contribution to the variance, so a fixed number of noise vectors should give a smaller error than Gaussian noise whenever the off-diagonal term is comparable.
In lattice QCD the method makes near-diagonal quark-loop traces, the chiral condensate $�langle�bar�Psi�Psi�rangle=�rm Tr}M^{-1}/V$, and the fermion-method topological susceptibility practical on a $16^3�times24$ lattice, with the paper quoting a computing-time saving of $5838$ over full inversion.
The same variance argument predicts that disconnected three-point insertions, such as the $�pi N$ sigma term and the flavor-singlet axial charge, are more tractable than disconnected two-point functions because only one quark loop is correlated with the nucleon propagator.
Because the trace is the best-estimated quantity, the authors propose using the stochastic trace to compute determinants, eigenvalue densities, and eigenvectors through the poles of $�rm Tr}�,G(�lambda+i�epsilon)$.
The method is explicitly limited to matrices whose inverses are dominated by diagonal and near-diagonal elements, since far off-diagonal signals decay exponentially while the stochastic error remains roughly constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $C_2=0$ mechanism is not specific to lattice QCD: it is a statement about Rademacher ($�pm1$) probes, so the same gain should appear in any trace estimator for a sparse or diagonally dominated matrix, and that is a cheap check on synthetic matrices.
The paper's $2.4$ factor is a single-point estimate; a natural extension is to map the ratio $r=�sum_{k�ne j}[M^{-1}_{kj}]^2/[M^{-1}_{jj}]^2$ and derive when the $Z_2$ advantage is large, since the variance formula makes the crossover explicit.
The stated failure for far off-diagonal propagators suggests a hybrid strategy: use $Z_2$ stochastic estimation for the near-diagonal part and deterministic solves for the few long-distance columns that carry the exponentially small signal.
The proposal to extract eigenvalues from poles of the stochastic trace could be tested by comparing the smeared density of states from $Z_2$ noise against a direct Lanczos or eigenvalue calculation on a small lattice.

Original abstract

We introduce a $Z_2$ noise for the stochastic estimation of matrix inversion and discuss its superiority over other noises including the Gaussian noise. This algorithm is applied to the calculation of quark loops in lattice quantum chromodynamics that involves diagonal and off-diagonal traces of the inverse matrix. We will point out its usefulness in its applications to estimating determinants, eigenvalues, and eigenvectors, as well as its limitations based on the structure of the inverse matrix.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We introduce a $Z_2$ noise for the stochastic estimation of matrix inversion and discuss its superiority over other noises including the Gaussian noise. This algorithm is applied to the calculation of quark loops in lattice quantum chromodynamics that involves diagonal and off-diagonal traces of the inverse matrix. We will point out its usefulness in its applications to estimating determinants, eigenvalues, and eigenvectors, as well as its limitations based on the structure of the inverse matrix.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce a $Z_2$ noise for the stochastic estimation of matrix inversion and discuss its superiority over other noises including the Gaussian noise.	contribution 1.00	0.93	0.61	asserted 1.00
s1	This algorithm is applied to the calculation of quark loops in lattice quantum chromodynamics that involves diagonal and off-diagonal traces of the inverse matrix.	method 0.60	0.82	0.05	asserted 0.99
s2	We will point out its usefulness in its applications to estimating determinants, eigenvalues, and eigenvectors, as well as its limitations based on the structure of the inverse matrix.	limitation 0.56	0.82	0.22	asserted 0.86

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.86.

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References

Works cited (12)

[1]

R. Scalettar, D. Scalapino, and R. Sugar, Phys. Rev. B34, 7911 (1986); S. Duane and J.B. Kogut, Nucl. Phys. B275 395, (1986); G.G. Bartrouni et al., Phys. Rev. D32, 2736 (1985); S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken, and R . Sugar, ibid. 35, 3972 (1987)

View on Pith 1986
[2]

K. Bitar et al, Nucl. Phys. B313 348, (1989); H.R. Fiebig and R.M. Woloshyn, Phys. Rev. D42 3520, (1990)

View on Pith 1989
[9]

S. Bernardson, P. McCarty, and C. Thron, King College Preprint KC/93-01

View on Pith
[3]

J. Smit and J. Vink, Nucl. Phys. B284 (1987) 234

View on Pith 1987
[4]

See, for example, P.C. Hohenberg and B. Halperin, Rev. Mod. Phy s. 49, 435 (1977). 5

View on Pith 1977
[5]

E. Nelson, Phys. Rev. 150, 1079 (1966)

View on Pith 1966
[6]

G. Parisi and Wu Yongshi, Scientia Sinica 24, 483 (1981)

View on Pith 1981
[7]

C. Thron, King College Preprint KC/92-01

View on Pith
[8]

S.J. Dong and K.F. Liu, Nucl. Phys. B26(Proc. Suppl.), 353 (1992)

View on Pith 1992
[10]

S.J. Dong and K.F. Liu, Nucl. Phys. B30(Proc. Suppl.), 487 (1993)

View on Pith 1993
[11]

S.J. Dong and K.F. Liu, under preparation

View on Pith
[12]

C. Wongtawatnugool and S.Y. Wu, J. Math. Phys. 22(4), 633 (1981). Figure Caption Fig. 1 The ﬁrst and second columns show the accumulated averages of the real and imaginary parts of various current loops as functions of the noise c onﬁgurations L for a gauge conﬁguration. The third and last columns give the correspo nding histograms for their distributions. 6

View on Pith 1981

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How the literature uses it

36 arXiv papers cite this in 41 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 10 sentences without a confident label

Uses its method or tool 26 papers

Since a naive calculation is prohibitively time consuming, we construct the all-to-all propagator by using low-lying modes of the overlap-Dirac operator [ref] and the stochastic noise method [ref] .

Chiral behavior of $K \to \pi l \nu$ decay form factors in lattice QCD with exact chiral symmetry · Calculation of meson correlators · 1.00
For the disconnected diagrams, we use spin-diluted stochastic sources [ref] , with $Z_2 ×Z_2$ noise [ref] , and perform the measurements on $n_src=64$ stochastic samples.

Mixing between flavor singlets in lattice gauge theories coupled to matter fields in multiple representations · Numerical Strategy · 1.00
The disconnected propagators $Sx,x$ of eqn:Gpiz require a different method to conventional point-to-all propagators; here we use stochastic estimation of the matrix inverse [ref] .

Disconnected contributions to the magnetic polarisability of the neutral pion · Propagator Methods · 1.00

Cites it as background 2 papers

In the context of disconnected diagrams or distillation, the effectiveness of different types of noise and variance reduction techniques has been studied extensively [ref] .

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A milestone for solving the second problem was the introduction of random-noise estimators [ref] which allow one to sum over many or all source points stochastically.

Frequency-splitting estimators of single-propagator traces · Conclusions · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

R. Scalettar, D. Scalapino, and R. Sugar, Phys. Rev. B34, 7911 (1986); S. Duane and J.B. Kogut, Nucl. Phys. B275 395, (1986); G.G. Bartrouni et al., Phys. Rev. D32, 2736 (1985); S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken, and R . Sugar, ibid. 35, 3972 (1987)

View on Pith 1986

[2] [2]

K. Bitar et al, Nucl. Phys. B313 348, (1989); H.R. Fiebig and R.M. Woloshyn, Phys. Rev. D42 3520, (1990)

View on Pith 1989

[3] [9]

S. Bernardson, P. McCarty, and C. Thron, King College Preprint KC/93-01

View on Pith

[4] [3]

J. Smit and J. Vink, Nucl. Phys. B284 (1987) 234

View on Pith 1987

[5] [4]

See, for example, P.C. Hohenberg and B. Halperin, Rev. Mod. Phy s. 49, 435 (1977). 5

View on Pith 1977

[6] [5]

E. Nelson, Phys. Rev. 150, 1079 (1966)

View on Pith 1966

[7] [6]

G. Parisi and Wu Yongshi, Scientia Sinica 24, 483 (1981)

View on Pith 1981

[8] [7]

C. Thron, King College Preprint KC/92-01

View on Pith

[9] [8]

S.J. Dong and K.F. Liu, Nucl. Phys. B26(Proc. Suppl.), 353 (1992)

View on Pith 1992

[10] [10]

S.J. Dong and K.F. Liu, Nucl. Phys. B30(Proc. Suppl.), 487 (1993)

View on Pith 1993

[11] [11]

S.J. Dong and K.F. Liu, under preparation

View on Pith

[12] [12]

C. Wongtawatnugool and S.Y. Wu, J. Math. Phys. 22(4), 633 (1981). Figure Caption Fig. 1 The ﬁrst and second columns show the accumulated averages of the real and imaginary parts of various current loops as functions of the noise c onﬁgurations L for a gauge conﬁguration. The third and last columns give the correspo nding histograms for their distributions. 6

View on Pith 1981