Local Sign Decomposition

• Local sign decomposition is a method that reformulates globally defined sign information into locally computable, indexed components.
• It facilitates tractability and canonical representation across diverse areas such as quantum Monte Carlo, financial modeling, and algebraic decompositions.
• By partitioning sign structures locally, the approach yields practical surrogates and precise decompositions for complex global phenomena.

Searching arXiv for papers related to local sign decomposition and adjacent usages of the term. Local sign decomposition denotes a family of constructions in which sign information that is globally defined, combinatorially coupled, or otherwise nonlocal is reorganized into locally indexed components, local surrogates, or local equivalence classes. The phrase is not attached to a single universal formalism. In recent arXiv usage it appears in several distinct technical settings: local basis optimization for mitigating the negative sign problem in quantum Monte Carlo (Murota et al., 30 Jan 2025); state-dependent sign–magnitude factorization of financial returns (Brou et al., 2 Jun 2026); local splitting of Loday and Courant algebroids (Lima, 3 May 2026); sign-cone decompositions of symmetric matrices by leading principal minors (Khare et al., 31 Jul 2025); phase-based local feature analysis via complex wavelet signs (Storath et al., 2012); sign-sensitive canonical decompositions in bidirected graphs (Kita, 2017); local cohomological plus/minus decompositions for symplectic self-dual Galois representations (Burungale et al., 25 Aug 2025); and sign-decompositions of torsion classes for radical-square-zero algebras (Aoki, 2018). Across these contexts, the common structure is not an exact shared definition but a recurring methodological pattern: a sign-bearing object is partitioned, factorized, or approximated using locally defined data, often to obtain tractability, canonical structure, or geometric interpretability.

1. Conceptual scope and recurrent pattern

A precise cross-disciplinary definition is not standard. What recurs is a shift from a single global sign object to a collection of local sign-bearing pieces, local sign constraints, or locally computable surrogates. In some settings this is an exact decomposition; in others it is explicitly only a surrogate or upper bound.

In quantum Monte Carlo for local spin Hamiltonians, the relevant global quantity is the sign severity of the partition-function expansion, measured by the negativity

$$\eta = \sqrt{ \frac{\langle |S|^2\rangle_{\mathrm v}}{|\langle S\rangle_{\mathrm v}|^2} }.$$

The paper “Local Basis Transformation to Mitigate Negative Sign Problems” does not use the phrase “local sign decomposition,” but it replaces the global low-temperature objective

$\lambda^+(G)-\lambda(G)$

by a sum of local bondwise penalties through the $L1$ adaptive loss, making local sign decomposition an optimization surrogate rather than an exact additive identity (Murota et al., 30 Jan 2025).

In other areas the decomposition is literal. For financial returns, one has the exact identity

$R_t=\operatorname{sign}(R_t)\,|R_t|$

or equivalently

$$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$

and the local aspect arises because the sign is modeled conditionally on contemporaneous magnitude, so sign behavior is “local in magnitude” (Brou et al., 2 Jun 2026). For symmetric matrices, the dense cone $LPM_n$ is partitioned into disjoint sign cones $LPM_n(\epsilon)$ indexed by the signs of leading principal minors, yielding a canonical signed Cholesky theory (Khare et al., 31 Jul 2025). For bidirected graphs, the nontrivial decomposition inside a circular component is governed by the existence or nonexistence of $(\alpha,\alpha)$-ditrails, so the sign-locality lies in endpoint sign constraints (Kita, 2017).

This suggests a useful “Editor’s term”: local sign architecture. In this sense, local sign decomposition is the organization of a sign-sensitive problem by local cells, local residues, local basis rotations, local endpoint signs, local states, or local minors. The exact mathematical content depends on the domain.

2. Local surrogates for global sign structure in quantum Monte Carlo

The most direct contemporary use of the idea in many-body physics is the local-basis framework of “Local Basis Transformation to Mitigate Negative Sign Problems” (Murota et al., 30 Jan 2025). The setting is stochastic series expansion / worldline quantum Monte Carlo for local quantum spin systems. The Hamiltonian is decomposed as

$$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$

and the partition function is written

$$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$

The sign problem is representation dependent: in the spin-language convention used in the paper, positive off-diagonal elements of the Hamiltonian matrix generate negative weights in QMC. The global sign-severity measure is the negativity

$\lambda^+(G)-\lambda(G)$0

and with the optimal virtual Hamiltonian $\lambda^+(G)-\lambda(G)$1, where off-diagonal matrix elements are replaced by their absolute values, one obtains

$\lambda^+(G)-\lambda(G)$2

At low temperature, if $\lambda^+(G)-\lambda(G)$3 and $\lambda^+(G)-\lambda(G)$4 have unique ground states,

$\lambda^+(G)-\lambda(G)$5

The local-basis ansatz is

$\lambda^+(G)-\lambda(G)$6

The key local object is the $\lambda^+(G)-\lambda(G)$7 adaptive loss

$\lambda^+(G)-\lambda(G)$8

This quantity is locally defined, bondwise, and is the paper’s closest formal analogue of local sign decomposition.

A central limitation is exact and structural: $\lambda^+(G)-\lambda(G)$9 Accordingly, the sign problem is not exactly additive over local pieces. The paper emphasizes that the local formalism is therefore not an exact decomposition of global negativity into independent bond contributions. Rather, in frustration-free systems it yields an upper bound. If $L1$0 is frustration-free, then

$L1$1

hence

$L1$2

The strongest evidence comes from random frustration-free parent Hamiltonians of MPS and PEPS, where optimizing $L1$3 significantly reduces gap negativity $L1$4, with near-linear correlation between reduction in $L1$5 and reduction in $L1$6 in the 1D random frustration-free MPS tests (Murota et al., 30 Jan 2025).

The same framework also recovers or nearly recovers known basis cures in the $L1$7-$L1$8-$L1$9-$R_t=\operatorname{sign}(R_t)\,|R_t|$0 chain, the spin-1 bilinear-biquadratic chain, and the Shastry–Sutherland model, while giving only partial mitigation in the kagome Heisenberg model. The paper further shows that allowing local unitary transformations can outperform orthogonal ones; in the kagome model, unitary optimization restored the symmetry under exchange $R_t=\operatorname{sign}(R_t)\,|R_t|$1 that orthogonal optimization broke (Murota et al., 30 Jan 2025). The local sign decomposition here is therefore best understood as a locally computable surrogate objective for basis engineering, especially justified in frustration-free or near-frustration-free settings.

3. Exact sign–magnitude decompositions and state-local sign models

A different meaning appears in time-series modeling of financial returns. In “A new decomposition approach to modeling financial returns: Conditioning sign on magnitude,” returns are decomposed exactly as

$R_t=\operatorname{sign}(R_t)\,|R_t|$2

or, in binary-coded form,

$R_t=\operatorname{sign}(R_t)\,|R_t|$3

There is no information loss in this representation (Brou et al., 2 Jun 2026).

The local aspect is not spatial but state-dependent. The proposed CSM model factors the joint law of magnitude and sign as

$R_t=\operatorname{sign}(R_t)\,|R_t|$4

with magnitude modeled by a multiplicative error model

$R_t=\operatorname{sign}(R_t)\,|R_t|$5

and sign modeled conditionally on the contemporaneous magnitude through a probit

$R_t=\operatorname{sign}(R_t)\,|R_t|$6

The expected return is then induced by integrating a magnitude-dependent sign probability against the conditional magnitude distribution: $R_t=\operatorname{sign}(R_t)\,|R_t|$7

The paper explicitly characterizes this as state-dependent or “local in magnitude” rather than local smoothing in the nonparametric sense. In empirical forecasting of monthly U.S. stock market excess returns, all models are estimated in a rolling window of length $R_t=\operatorname{sign}(R_t)\,|R_t|$8, producing one-step-ahead forecasts over 487 out-of-sample months, and predictor subsets of size $R_t=\operatorname{sign}(R_t)\,|R_t|$9 are chosen in-sample mainly by maximizing AUC (Brou et al., 2 Jun 2026). Under absolute loss, the CSM baseline yields $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$0 values of $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$1, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$2, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$3, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$4, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$5, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$6, $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$7, and $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$8 across $$R_t=M_t(2S_t-1), \qquad M_t=|R_t|\ge 0, \qquad S_t=\mathbf 1\{R_t>0\},$$9. The paper is careful, however, not to claim a monotone sign-on-magnitude effect; the evidence for the channel is indirect but substantial, coming from component correlations, forecast performance, and competitiveness against copula-based return-decomposition methods (Brou et al., 2 Jun 2026).

This illustrates a broader point: local sign decomposition may refer not only to partitioning a sign structure into spatial or combinatorial pieces, but also to conditioning directional behavior on a local state variable.

4. Local splitting, canonical partitions, and exact sign-indexed decompositions

Several papers use genuinely exact sign-based decompositions, but in algebraic, geometric, combinatorial, or arithmetic senses rather than in statistical mechanics.

In linear algebra, “Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices” defines

$LPM_n$0

for a sign pattern $LPM_n$1, where $LPM_n$2 is the leading principal $LPM_n$3 submatrix. The dense cone

$LPM_n$4

is thus partitioned by local sign data of leading principal minors (Khare et al., 31 Jul 2025). On each cone one has a canonical signed Cholesky factorization

$LPM_n$5

where $LPM_n$6 is the diagonal sign matrix determined by $LPM_n$7, and $LPM_n$8 is lower triangular with positive diagonal. More generally, for any fixed $LPM_n$9,

$LPM_n(\epsilon)$0

with $LPM_n(\epsilon)$1 unique. The decomposition by leading-principal-minor signs is finer than inertia and coarser than prescribing all principal minors, and it is exactly the structure needed for the generalized Cholesky theory (Khare et al., 31 Jul 2025).

In representation theory of finite-dimensional algebras, Aoki’s “Classifying torsion classes for algebras with radical square zero via sign decomposition” decomposes the lattice of torsion classes into $LPM_n(\epsilon)$2 sign strata indexed by $LPM_n(\epsilon)$3,

$LPM_n(\epsilon)$4

For $LPM_n(\epsilon)$5, each stratum is controlled by a hereditary algebra $LPM_n(\epsilon)$6, and there are isomorphisms of posets

$LPM_n(\epsilon)$7

At the support $LPM_n(\epsilon)$8-tilting level,

$LPM_n(\epsilon)$9

Here the sign vector is local in the sense that each coordinate controls a simple module or primitive idempotent separately (Aoki, 2018).

In combinatorial representation theory, Tan and Teo’s “Sign Sequences and Decomposition Numbers” attaches to a partition $(\alpha,\alpha)$0 and residue $(\alpha,\alpha)$1 the sign sequence

$(\alpha,\alpha)$2

where removable and indent $(\alpha,\alpha)$3-nodes are ordered by horizontal position. The main theorem gives the exact formula

$(\alpha,\alpha)$4

Thus a decomposition number is determined by same-residue local sign-sequence combinatorics and well-nested latticed paths (Tan et al., 2011).

In bidirected graph theory, the relevant decomposition is the signed general Kotzig–Lovász decomposition. A bidirected graph assigns a sign $(\alpha,\alpha)$5 or $(\alpha,\alpha)$6 to each end of each edge. For each $(\alpha,\alpha)$7, one defines $(\alpha,\alpha)$8 if either $(\alpha,\alpha)$9, or $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$0 and $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$1 are circularly connected and there is no $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$2-ditrail between them. The main theorem proves that $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$3 is an equivalence relation, giving a canonical sign-sensitive partition inside each circular component (Kita, 2017). In digraphs the decomposition is trivial because same-sign ditrails do not occur; the nontrivial structure is characteristic of genuinely bidirected graphs.

5. Local sign decomposition in geometry, topology, and arithmetic

In differential-geometric and algebroid settings, local decomposition refers to splitting local structure rather than extracting sign from scalar quantities.

For general Loday algebroids, “Local decomposition and linearization of Loday brackets” studies local splitting-type results and proves local splitting theorems for Courant algebroids (Lima, 3 May 2026). The fundamental local normal form is a product decomposition

$$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$4

near a point where the anchor is nonzero, with

$$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$5

In adapted coordinates $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$6 and frame $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$7, the local bracket structure becomes

$$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$8

so the structure functions depend only on the transverse variables. The paper studies local splitting/decomposition of Loday brackets, not signs in the scalar sense, but it is one of the clearest cases in which “local decomposition” is the canonical term (Lima, 3 May 2026).

In complex geometry, “Local structure of closed symmetric 2-differentials” shows that a closed symmetric $$H=\sum_{b\in\mathcal B} h_b, \qquad G\equiv -H, \qquad g_b\equiv -h_b,$$9-differential $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$0 of rank $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$1 on a complex surface is locally a product of closed holomorphic $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$2-differentials away from the breakdown locus $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$3, and near a general point of an irreducible component $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$4 has the local normal form

$$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$5

This yields a generalized local factorization into closed $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$6-differentials, possibly multivalued and singular, with the poles of the meromorphic exponents bounded by the order of contact $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$7 of the two foliations (Bogomolov et al., 2014). This is a decomposition theory for local differential structure rather than sign per se, but it occupies the same methodological niche of replacing a global object by controlled local factors.

In arithmetic geometry, “A local sign decomposition for symplectic self-dual Galois representations of rank two” gives a literal plus/minus decomposition

$$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$8

for generic symplectic self-dual rank-two $$Z=\mathrm{Tr}[e^{\beta G}] = \sum_{c\in\mathcal C} W(c).$$9-adic representations of $\lambda^+(G)-\lambda(G)$00, with $\lambda^+(G)-\lambda(G)$01 free rank-one Lagrangian submodules (Burungale et al., 25 Aug 2025). The sign is governed by the completed epsilon constant

$\lambda^+(G)-\lambda(G)$02

and for de Rham specializations one has

$\lambda^+(G)-\lambda(G)$03

The decomposition is functorial, base-change compatible, and characterized uniquely by these properties. Here local sign decomposition means a canonical signed polarization of local Galois cohomology.

Finally, in Banach–Stone-type geometry, “Globalization of local sign structures for phase-isometries on uniform algebras” studies surjective phase-isometries $\lambda^+(G)-\lambda(G)$04 that preserve maximal convex subsets only up to sign. For each boundary point $\lambda^+(G)-\lambda(G)$05, one has local sign ambiguity in

$\lambda^+(G)-\lambda(G)$06

and the paper’s main contribution is to globalize the residual local signs into a single map

$\lambda^+(G)-\lambda(G)$07

obtaining a boundary representation by a global sign function, a unimodular weight, a boundary homeomorphism, and a clopen decomposition into complex-linear and conjugate-linear parts (Enami et al., 21 Jun 2026). This is not a decomposition of a sign-bearing scalar field, but a globalization theorem for local sign structure.

6. Signal-processing, optimization, and computational interpretations

In signal analysis, the most direct local sign object is the complex-wavelet signature of a signal. In “Signal Analysis based on Complex Wavelet Signs,” the signature is defined by

$\lambda^+(G)-\lambda(G)$08

if the limit exists, and $\lambda^+(G)-\lambda(G)$09 otherwise (Storath et al., 2012). Here

$\lambda^+(G)-\lambda(G)$10

for $\lambda^+(G)-\lambda(G)$11. The signature is zero at sufficiently regular points and nonzero at salient singularities. Jumps yield purely imaginary signatures,

$\lambda^+(G)-\lambda(G)$12

symmetric cusps yield purely real signatures,

$\lambda^+(G)-\lambda(G)$13

and one-sided cusps yield

$\lambda^+(G)-\lambda(G)$14

Thus local signal morphology is decomposed by asymptotic phase orientation in the complex plane. The paper explicitly interprets the orientation of the signature as an indicator of local symmetry or antisymmetry. This is a genuinely local sign-based decomposition of signal structure (Storath et al., 2012).

In social-network analysis, “Statistical Link Label Modeling for Sign Prediction” does not speak of local sign decomposition, but its probabilistic structure is one. It models the sign of an edge through local source/target/peer contexts and interpolates local estimators with cluster-level global ones using support-sensitive smoothing. The paper’s central claim is a dilemma between local and global structures, resolved by smoothing; local evidence is strong when dense but must be backed off to global structure under sparsity (Javari et al., 2018). This is a decomposition of sign evidence into local generators and global fallback.

In symbolic computation, “Cylindrical Algebraic Decomposition Using Local Projections” replaces a global sign-invariant CAD projection phase by cell-wise local projection sets (Strzebonski, 2014). A local projection sequence $\lambda^+(G)-\lambda(G)$15 for $\lambda^+(G)-\lambda(G)$16 at $\lambda^+(G)-\lambda(G)$17 is defined so that if lower-level projection factors keep constant sign on a cell containing $\lambda^+(G)-\lambda(G)$18, then the relevant higher-level polynomials are delineable over that cell. The algorithm $\lambda^+(G)-\lambda(G)$19 computes truth-invariant local cells using only the polynomial signs needed around a given sample point and for a given branch. In the worked example

$\lambda^+(G)-\lambda(G)$20

global CAD constructs $\lambda^+(G)-\lambda(G)$21 cells, whereas $\lambda^+(G)-\lambda(G)$22 constructs $\lambda^+(G)-\lambda(G)$23 cells because $\lambda^+(G)-\lambda(G)$24 is only needed on a much smaller locus (Strzebonski, 2014). This is not a sign decomposition in the matrix- or graph-theoretic sense, but it is a localized sign-control strategy.

6. Cross-domain themes, misconceptions, and limits

The term should not be treated as denoting one mature universal theory. A common misconception is to assume that “local sign decomposition” always means an exact decomposition of a global sign quantity into independent local terms. The quantum Monte Carlo case provides a direct counterexample: $\lambda^+(G)-\lambda(G)$25 so the local object is only a surrogate objective, albeit a useful one (Murota et al., 30 Jan 2025). By contrast, the sign–magnitude factorization of returns and the partition of $\lambda^+(G)-\lambda(G)$26 by signs of leading principal minors are exact decompositions (Brou et al., 2 Jun 2026, Khare et al., 31 Jul 2025).

A second misconception is to equate locality with nonparametric smoothing. In the CSM return model, locality means conditioning on the magnitude state, not kernel-local regression (Brou et al., 2 Jun 2026). In bidirected graphs, locality lies in endpoint sign type and internal sign-switching constraints, not in graph distance (Kita, 2017). In Galois cohomology, locality means local $\lambda^+(G)-\lambda(G)$27-adic Galois representations and local Tate pairings, not spatial localization (Burungale et al., 25 Aug 2025).

A third misconception is that sign decompositions are always purely combinatorial. The cited work shows several distinct mechanisms: basis changes in Hilbert space (Murota et al., 30 Jan 2025), geometric splitting theorems (Lima, 3 May 2026), phase limits of wavelet coefficients (Storath et al., 2012), cohomological involutions defined via epsilon-isomorphisms and $\lambda^+(G)-\lambda(G)$28-modules (Burungale et al., 25 Aug 2025), and support-adaptive statistical interpolation (Javari et al., 2018).

Despite these differences, several structural themes recur. One is tractability via localization: replacing an exponentially hard or globally entangled object by local surrogates, local factor models, or local projection sets. Another is canonical structure via sign indexing: decomposing an object into chambers, cones, orthants, or equivalence classes labeled by sign data. A third is compatibility with ambient geometry: local signs often encode symplectic, Riemannian, wavelet-phase, or foliation-adapted structure rather than mere Boolean polarity.

This suggests that local sign decomposition is best regarded as a family resemblance concept. It names a class of techniques in which sign information is localized, indexed, or factorized so that a problem becomes structurally transparent or computationally manageable. The specific mathematical content must therefore be read from the domain-specific formalism rather than inferred from the phrase alone.