Inverse fitting from separator observations

Consider a power diagram with fixed sites and unknown weights. Suppose that the available data do not describe complete cells. Instead, they specify where the separator should lie for selected pairs of sites. The inverse problem is to find one set of site weights that reconciles these local observations.

The central reduction is simple: the position of one pairwise separator determines one difference of weights. A collection of observations therefore becomes a graph of desired weight differences. Exact compatibility is a cycle condition on that graph; noisy fitting leads to a gauge-aware graph-Laplacian problem. Geometric realization remains a separate question because a third site may block a pair even when its requested separator equation is fitted exactly.

The current Python API stores these data in SeparatorObservations. This page uses the same term separator observations while stating the main mathematical ideas independently of API details.

Observation model for one pair

Fix site \(i\) and one selected image \(q_j\) of site \(j\). In a nonperiodic domain, \(q_j=p_j\). Define

\[ \Delta_{ij}=q_j-p_i, \qquad d_{ij}=\lVert\Delta_{ij}\rVert>0, \]

and parameterize the connector line by

\[ x(t)=p_i+t\Delta_{ij}. \]

The pairwise separator meets this line where the two power distances agree. Solving the resulting scalar equation gives the normalized connector coordinate

\[ t_{ij}=\frac12+\frac{w_i-w_j}{2d_{ij}^2}. \]

The intersection always exists on the connector line, although it need not lie between the two sites. If the observation is the signed distance from site \(i\) along that line, write

\[ s_{ij}=d_{ij}t_{ij} =\frac{d_{ij}}2+\frac{w_i-w_j}{2d_{ij}}. \]

Both conventions, and any affine reparameterization of them, can be written as

\[ y_r=\beta_r+\alpha_r(w_{i_r}-w_{j_r}), \qquad \alpha_r\ne 0. \]

Thus observation \(r\) implies the desired weight difference

\[ z_r^{\mathrm{obs}} =\frac{y_r^{\mathrm{obs}}-\beta_r}{\alpha_r}. \]

The data are local, but the same site weight appears in every observation that touches that site. Global consistency is therefore a graph problem.

Compatibility on an observation graph

The simplest nontrivial example has three sites. If observations prescribe the differences \(z_{12}\), \(z_{23}\), and \(z_{31}\), then any global weight vector must satisfy

\[ z_{12}+z_{23}+z_{31}=0. \]

A nonzero sum means that the three local requirements cannot all come from one set of weights. The same principle holds on every cycle of a larger observation graph.

Formally, create one graph vertex per site and one oriented edge per observation. Parallel edges are retained, so repeated measurements and different periodic images of the same site pair remain distinguishable. Let \(B\) be the vertex-edge incidence matrix, with the column for observation \(r\) equal to \(e_{i_r}-e_{j_r}\). Exact compatibility is the linear system

\[ z^{\mathrm{obs}}=B^\mathsf{T}w. \]

The observations are compatible if and only if their signed sum vanishes around every cycle of the observation multigraph. Equivalently, \(z^{\mathrm{obs}}\) lies in the image of \(B^\mathsf{T}\), or is orthogonal to the graph cycle space.

Keeping parallel edges is mathematically important. In a periodic problem, two observations may refer to different images of the same site pair. They still constrain the same weight difference, and together they form a two-edge cycle whose consistency would be hidden by a simple unlabelled pair set.

Identifiability, global gauge, and component offsets

Only observations with positive confidence contribute to the objective and identify differences. These observations define the effective observation graph.

A zero-confidence observation is still a distinct row of the resolved data and a distinct column of the full incidence matrix. Its effective weight is zero, however, so it contributes neither to the Laplacian nor to its right-hand side and cannot connect effective components.

Within one connected effective component, all relative weights are determined by exact compatible data. One additive constant per component remains unobserved. When the effective graph is connected, that constant is exactly the global additive gauge of the complete power diagram.

When the effective graph has several components, the interpretation is more subtle. One common shift of all sites is still harmless geometric gauge, but the additional relative offsets between components are not determined by the separator data and may change the full tessellation. A solver may choose a canonical centering or align components to a reference, but that choice is a policy or prior, not information recovered from the disconnected observations.

Noisy observations and the Laplacian estimator

Let \(\lambda_r\ge 0\) be the confidence assigned to observation \(r\). The quadratic estimator minimizes measurement-space mismatch:

\[ \min_w \frac12\sum_r \lambda_r\left(y_r^{\mathrm{obs}}- \beta_r-\alpha_r(w_{i_r}-w_{j_r})\right)^2. \]

Define

\[ \rho_r=\lambda_r\alpha_r^2, \qquad q_r=\lambda_r\alpha_r(y_r^{\mathrm{obs}}-\beta_r), \qquad R=\operatorname{diag}(\rho_r). \]

After converting observations to implied differences, the same problem is

\[ \min_w \frac12\left\|R^{1/2} (B^\mathsf{T}w-z^{\mathrm{obs}})\right\|_2^2. \]

Its normal equations are

\[ Lw=Bq, \qquad L=BRB^\mathsf{T}. \]

The equality \(q=Rz^{\mathrm{obs}}\) is algebraically valid, but numerical implementations construct \(q_r\) directly. An implied difference may overflow even when \(\lambda_r\alpha_r(y_r^{\mathrm{obs}}-\beta_r)\) is finite, so z_obs is diagnostic data rather than a required normal-equation intermediate.

The matrix \(L\) is the weighted graph Laplacian of the effective observation graph. Its nullspace consists of vectors that are constant on each connected component, so one explicit alignment condition is required per component. After those conditions are chosen, the quadratic fit is unique.

The fitted edge differences are the weighted projection of the observed edge data onto the space of compatible differences. This interpretation separates two useful diagnostics:

measurement-space residuals, expressed in the units of the observed separator positions;
difference-space residuals, expressed as errors in the implied values of \(w_i-w_j\).

They are related through \(\alpha_r\), but they are not numerically interchangeable and should not share an ambiguous summary name.

With L2 strength \(\lambda\ge 0\) and reference \(w^{\mathrm{ref}}\), add

\[ \frac{\lambda}{2}\lVert w-w^{\mathrm{ref}}\rVert_2^2 \]

under the one-half objective convention used above. The regularized normal system is

\[ Aw=b, \qquad A=L+\lambda I, \qquad b=Bq+\lambda w^{\mathrm{ref}}. \]

The one-half factors are part of the reported objective contract; they are not only a normal-equation convenience. Positive L2 regularization removes the component-constant null directions from \(A\), but it does not connect the observation graph or turn prior-selected component offsets into information identified by the separator data.

Robust losses and regularization

The graph structure does not depend on squared loss. Convex edge-separable losses, such as the Huber loss, can reduce the influence of outlying separator observations while retaining the same site-difference model.

Regularization may stabilize weakly informed problems or choose among unidentified component alignments. It also adds information to the estimation problem. A solution selected by a reference or penalty should therefore be reported as such, rather than described as uniquely identified by the separator observations alone.

For measurement residual

\[ e_r(w)=\beta_r+\alpha_r(w_{i_r}-w_{j_r})-y_r^{\mathrm{obs}}, \]

and confidence \(c_r\), the implemented mismatch contribution is \(\sum_r c_r\ell(e_r)\). Confidence multiplies only this mismatch term. It does not multiply scalar penalties or hard restrictions. The two mismatch losses are

\[ \ell_{\mathrm{sq}}(e)=\frac12e^2 \]

and

\[ \ell_\delta(e)= \begin{cases} \frac12e^2, & |e|\le\delta,\\ \delta\left(|e|-\frac12\delta\right), & |e|>\delta. \end{cases} \]

Thus a Huber model with every residual in the quadratic branch is exactly the squared-loss model. L2 regularization is \(\frac{\lambda}{2}\lVert w-w^{\mathrm{ref}}\rVert_2^2\), with gradient \(\lambda(w-w^{\mathrm{ref}})\) and Hessian contribution \(\lambda I\). A reference with \(\lambda=0\) contributes nothing to the objective, although the documented output alignment policy may use it to choose otherwise unidentified component offsets.

Evaluation preserves the same formulas at extreme binary64 scales. A zero-confidence row returns exact zeros before its residual is formed. Positive confidence is applied in a scale-safe order before squaring. In particular, the quadratic scale uses \(\sqrt{c_r}\sqrt{1/2}\), rather than first halving \(c_r\), so the minimum positive binary64 confidence remains effective. Huber evaluates only the active branch. Zero L2 strength likewise returns zero before subtracting the reference; positive L2 scales differences before a stable sum-of-squares reduction. These rules preserve finite weighted values without clipping genuinely non-representable results.

The residual is evaluated directly as the complete affine combination \(\beta+\alpha w_i-\alpha w_j-y^{\mathrm{obs}}\). It is not reconstructed by subtracting the target from a stored prediction. Consequently, a prediction that lies outside binary64 range may be non-finite while a cancellation with the target leaves a finite, reportable residual and mismatch value. Mismatch evaluation also applies confidence and the loss scale directly to the complete affine combination, so a finite weighted objective does not require the unscaled residual itself to fit binary64. The Huber linear expression is evaluated as one scaled combination rather than as the difference of two potentially overflowing terms. Ordinary rows use vectorized arithmetic; rows whose forward-error bound exceeds \(64\epsilon_{64}\) times their result magnitude use vectorized error-free product/sum compensation. Every raw compensated product must have normal-range exponent margin; otherwise the row takes the exact binary64-input fallback before that product is executed. The ordinary pass reuses its affine products and one forward-error calculation. A representable compensated residual and its squared mismatch share the same rounded residual, while exact exceptional rows and non-representable residuals form the objective from the exact affine operands. This conditioning criterion resolves the same relative scale as the hard-row policy and preserves finite destructive cancellation before errors become material to that policy.

The existing soft-interval strength \(s\) means

\[ s\left(\max(a-y,0)^2+\max(y-b,0)^2\right), \]

and the existing exponential-boundary strength means

\[ s\left[ \exp\left(\frac{a+m-y}{\tau}\right) +\exp\left(\frac{y-(b-m)}{\tau}\right) \right]. \]

These terms do not acquire a one-half factor. For reciprocal repulsion, define the contribution from one inward boundary distance \(d\) as

\[ q(d)= \begin{cases} 0, & d\ge m,\\ s\left(\dfrac1d-\dfrac1m\right), & \epsilon<d<m,\\ s\left[ \left(\dfrac1\epsilon-\dfrac1m\right) -\dfrac{d-\epsilon}{\epsilon^2} \right], & d\le\epsilon. \end{cases} \]

The full interval contribution is \(q(y-a)+q(b-y)\). The tangent continuation is finite and convex below \(\epsilon\). Value and first derivative are continuous at \(\epsilon\), where the linear branch applies. At \(m\), the inactive branch applies: value is continuous, but the derivative jumps, so the activation point has no unique derivative.

Value, first derivative, and second derivative share the same branch classification but are evaluated separately. Scale-safe reciprocal products avoid first forming \(d^2\), \(d^3\), or \(\epsilon^2\), including for valid very small parameters.

For a positive soft-interval strength, the active-boundary displacement is scaled before its square is formed, and its first derivative is evaluated as a scaled difference. Reciprocal quotient differences and continuation terms are likewise combined before binary64 range conversion. These evaluation choices preserve the formulas above; they do not clip true extended-real results. Objective evaluation derives soft and reciprocal boundary distances directly from the complete affine prediction formula and the boundary. It therefore does not require the separately stored public prediction to be representable. When a rounded inward distance is not safely separated from \(\epsilon\) or \(m\), reciprocal branch selection uses the exact binary64 operands for scalar and complete-affine expressions. Thus rounding a distance onto a branch boundary cannot silently deactivate a value or its derivatives. The exact exceptional conversion for an objective returns infinity on genuine range overflow. It is intentionally distinct from the finite saturation used only to approximate tolerance-expanded hard-bound endpoints.

A zero-strength scalar penalty is absent. Its value and derivatives are exactly zero without evaluating its exponential, reciprocal, or other branch expressions; it does not affect graph coupling, backend selection, or the quadratic operator.

A single fixed normal system represents squared measurement mismatch plus optional L2 regularization. Hard intervals and equalities restrict its feasible set but do not change the quadratic objective matrix; their constrained optimum need not satisfy the unconstrained normal equation. Huber mismatch and additional scalar penalties are not, in general, represented by the same fixed Laplacian system.

Numerically, the scaled weighted least-squares form is the canonical solver model. For one component, its observation rows are

\[ \sqrt{c_r}\,\alpha_r(e_{i_r}-e_{j_r})^\mathsf{T}w =\sqrt{c_r}(y_r^{\mathrm{obs}}-\beta_r), \]

and positive L2 adds the rows \(\sqrt{\lambda}I w=\sqrt{\lambda}w^{\mathrm{ref}}\). The direct augmented solve avoids explicitly squaring its condition number.

The implementation applies exact power-of-two scales to the component variables and objective rows before assembling the design. It also keeps the high and low parts of each target offset \(y_r^{\mathrm{obs}}-\beta_r\) as separate right-hand-side contributions. Consequently, a minimum-positive confidence or L2 strength is not erased merely because another row is much larger. A mathematically nonzero term that cannot be represented in the scaled binary64 system produces a structured numerical failure rather than a modified objective.

For an unregularized component, the internal gauge fixes the site with the largest scaled weighted degree and solves the remaining contrasts. The result is then shifted uniformly to the documented public gauge, so this conditioning choice does not change any predicted separator. Positive L2 removes the gauge mathematically; the implementation centers variables around the reference mean while retaining all site coordinates and the full \(\sqrt{\lambda}I\) block. It does not change one or two selected coordinates after solving in order to force an exact floating-point component sum.

The augmented design remains the authoritative representation. For ordinary components with a conservative design-condition bound comfortably below the binary64 normal-equation limit, the direct solver may try a guarded dense or sparse normal-system candidate for performance. It uses the same component scales and split right-hand side, and it is accepted only after the same source-objective and stationarity checks. Assembly, solve, or certification failure falls back to the unsquared augmented system.

Outside that guarded path, dense components use a QR solve with an SVD rank and condition estimate. High-condition designs also compare iterative refinement with a rank-revealing truncated candidate and retain truncation only when an accurately evaluated augmented residual is smaller. Sparse components solve the unsquared augmented problem through a sparse KKT factorization and retain the incidence structure without a dense site matrix. Neither candidate path changes linear backend at a site-count threshold. ADMM uses the selected dense or sparse prepared weight system for every iteration. Explicit ADMM on squared mismatch plus optional L2 still performs ADMM, and its final iterate must pass the common quadratic certificate.

A quadratic result is reported as successful only after certification. The final binary64 weights are checked after scale reversal and public gauge canonicalization. A floating objective equal to zero is insufficient: exact-zero proof requires every positive-confidence affine residual to vanish exactly and, under positive L2, every coordinate to equal its reference. Otherwise every guarded normal, dense augmented, sparse augmented, and final quadratic ADMM candidate needs a conservative forward objective-gap bound.

Positive L2 supplies the singular-value lower bound from its scaled \(\sqrt{\lambda}I\) block. Without L2, root a maximum-bottleneck spanning tree at the gauge anchor. The path structure of its inverse reduced incidence gives the conservative bound \(\sigma_{\min}\geq 1/\lVert T^{-1}\rVert_F\). The same proved identity/tree lower bound is used for normal, dense augmented, sparse augmented, and final quadratic ADMM candidates; dense SVD information remains a rank and condition diagnostic. Combining this lower bound with an upper bound on the source gradient norm gives [ f(w)-f(w^\star)\leq \frac{\lVert\nabla f(w)\rVert_2^2} {2\sigma_{\min}^2}. ] The gradient upper bound starts with ordinary binary64 forward-error accounting, tightens it with per-row extended-precision accounting when the platform provides genuinely wider arithmetic, and may finally use a bounded exact-dyadic accumulation. None of these paths changes the source objective. No factor path can report success without such a bound or an exact-zero proof. Bounded exact binary64-input helpers may improve a candidate or tighten its gradient bound, but are not an unbounded production matrix solver and do not change the meaning of success above a helper threshold. If a material weak term or the continuous optimum cannot be represented accurately by the public binary64 weight vector, the solver returns numerical_failure instead of a false optimal result.

For a bounded small component, exact rational arithmetic may compute both the continuous optimum and the exact source-objective gap of a binary64 candidate. This is a tighter implementation of the same continuous forward-gap certificate, not a change of optimization domain. Rounding each coordinate of the exact optimum independently does not in general minimize a coupled quadratic over the binary64 lattice. Accordingly, an exactly zero continuous optimum is successful only when the returned binary64 weights make every source-objective term exactly zero; otherwise the representability limitation is reported as numerical_failure.

The optional direct warm start for a nonquadratic ADMM problem is only an acceleration. Supported numerical failure falls back to the reference or zero seed. For Huber mismatch without scalar penalties or hard bounds, the solver also retains the best candidate according to the authoritative Huber-plus-L2 objective and never discards an exactly zero-objective candidate.

Well-conditioned normal right-hand sides, operator products, and mismatch-only weighted averages use vectorized arithmetic under the same \(64\epsilon_{64}\) relative forward-error target as complete affine evaluation. Nearly balanced opposite-sign averages, same-sign barycentric forms that lose a small weighted contribution, sufficiently imbalanced positive weights, normalized weights that would underflow, and mixed-sign incidence sums that fail the relative forward-error criterion use exact finite-input exceptional fallbacks. Product range checks use operand exponents to route potentially subnormal multiplication through exponent-scaled arithmetic without globally suppressing underflow. Complete sum-of-products evaluation also prevents a finite nonzero term that rounds to zero in isolation from disappearing before aggregation. The Huber linear proximal displacement similarly evaluates its confidence, threshold, and ADMM denominator as one ratio-product rather than first forming the reciprocal denominator. Huber proximal branch selection evaluates the unnormalized lower and upper optimality tests as complete sum-of-products expressions, avoiding comparisons with rounded target-plus-or-minus-threshold values. Exceptional tests retain their exact sign even when their nonzero magnitude lies below binary64 range. Direct Huber values and derivatives classify the complete residual against both threshold boundaries by the same exact-sign rule. Mean-absolute diagnostics use exponent-aware normalization and omit only ratios too small to affect the rounded result, without executing an underflowing division.

Certified scalar proximal coordinates

For a fixed ADMM measurement coordinate, the implemented scalar problem is

\[ \min_{a\le y\le b} c\,\ell(y-y^{\mathrm{obs}}) +\sum_p p(y)+\frac{\rho}{2}(y-v)^2, \qquad \rho>0. \]

The positive quadratic makes this objective strongly convex and coercive, so it has one continuous minimizer on bounded, equality, one-sided, and fully unbounded domains. The scalar solver represents the one-sided derivative as \([\hat g_-,\hat g_+]\). If \(S\) bounds the magnitude of its evaluated contributions and \(E\) is their floating evaluation allowance, it widens the interval to \(G_-=\hat g_--E\), \(G_+=\hat g_++E\) and uses

\[ \tau_{\mathrm{KKT}} =8\operatorname{ulp}(0)+64\epsilon_{64}S. \]

The interior residual is \(\max(G_-,-G_+,0)\). At a lower endpoint it is \(\max(-G_+,0)\), and at an upper endpoint it is \(\max(G_-,0)\). A successful point must put that residual within the non-configurable tolerance. Reciprocal margin activations use their convex subgradient intervals: \([-s/m^2,0]\) at the lower activation and \([0,s/m^2]\) at the upper activation.

Exact rational breakpoint locations cover Huber thresholds, soft boundaries, reciprocal epsilon and margin transitions, and finite hard bounds. If a real breakpoint is not binary64, both neighboring floats are checked. The solver first checks endpoints, the vectorized mismatch-only candidate, and those breakpoint candidates. It then maintains a certified negative-left, positive-right derivative bracket. Safeguarded Newton is confined to one smooth piece; otherwise ordered-binary64 bisection guarantees a new float. Exponential contributions use a common signed-log scale, so raw individual overflow does not turn a meaningful derivative sign into nan.

Success requires an equality certificate, a point-KKT certificate, or localization between adjacent feasible binary64 values with the required opposite derivative signs. The selected point, its feasible nextafter neighbors, finite bounds, and breakpoint candidates are compared through the authoritative objective before return. Reaching an expansion or iteration limit is never success. An uncertified coordinate becomes the existing structured numerical_failure, including row and last-certificate evidence; an attempted but failed proximal update is not counted as a completed ADMM iteration.

Hard interval and equality restrictions

An admissible interval in measurement space,

\[ a_r\le y_r\le b_r, \]

can be converted, with the orientation of \(\alpha_r\) taken into account, into a bound on one weight difference:

\[ \ell_r\le w_{i_r}-w_{j_r}\le u_r. \]

Each upper or lower bound is a directed difference inequality. The full system is feasible if and only if its directed constraint graph contains no negative cycle. When a negative cycle exists, it provides a compact contradiction witness that identifies a mutually incompatible subset of restrictions.

This graph precheck distinguishes genuine infeasibility from ordinary numerical non-convergence.

Floating-point hard-bound classification uses a scale-aware roundoff policy, separate from solver convergence tolerance. For lower bound \(a_r\), prediction \(y_r\), upper bound \(b_r\), and binary64 machine epsilon \(\epsilon_{64}\), define

\[ v_r=\max(a_r-y_r,\ y_r-b_r,\ 0), \]

\[ t_r=10^{-12} +64\epsilon_{64}\max(|a_r|,|y_r|,|b_r|). \]

The row is satisfied exactly when \(v_r\le t_r\). A row containing a non-finite bound or prediction is never satisfied.

For finite bounds, this predicate defines one tolerance-expanded closed interval in measurement space. The feasibility precheck derives that accepted measurement interval first and then maps its endpoints through \(y_r=\beta_r+\alpha_r(w_{i_r}-w_{j_r})\) into difference bounds. Bellman–Ford uses those bounds directly; it does not apply the measurement-space \(10^{-12}\) tolerance again to path distances measured in weight-difference units. Precheck feasibility and final classification therefore refer to the same accepted set. A constrained ADMM solve projects onto that same tolerance-expanded measurement interval, not the narrower unexpanded user interval. The original bounds still define raw violation and tolerance reporting. The relative tolerance product is evaluated scale-safely, so subnormal finite bounds do not raise merely because their relative contribution rounds away next to the absolute term.

Reports retain the maximum raw violation and separately expose the maximum tolerance actually used. A hard-constrained solver reports success only after both its primal/dual convergence tests and this row predicate pass.

Algebraic fitting and geometric realization

The affine observation law concerns the separator of two sites considered as a pair. It does not guarantee that the same pair is a boundary in the complete power diagram. A competitor can block the pair, one endpoint cell can be empty, or a different periodic image can be realized.

The computational workflow therefore has four distinct layers:

fixed-observation fit: reconcile the requested weight differences;
forward realization: compute the power diagram from the fitted weights;
boundary matching: compare requested pairs and image shifts with the realized boundaries;
optional outer refinement: revise the active observation set and refit.

A fit may be algebraically excellent and geometrically unsupported. Conversely, a realized pair may be absent from the observation set. Both outcomes are scientifically meaningful and should remain visible in the result.

Periodic image bookkeeping

A periodic observation refers to one selected image of the second site. The realization layer should therefore distinguish four outcomes:

the requested image is realized;
the same site pair is realized only through another image;
the pair is not realized;
one or both endpoint cells are empty.

Silently replacing the requested image by the nearest or currently realized image changes the observation. Such a change may be offered as an explicit repair strategy, but not as a hidden fallback.

Connector visibility and clearance

Full boundary realization is a global tessellation question. The manuscript also studies a narrower local diagnostic: whether the connector line passes through the relative interior of the pair face.

Let \(x_{ij}(w)\) be the connector-separator point. For each competitor \(k\ne i,j\), define its slack at that point by

\[ g_{ijk}(w)=\pi_k\bigl(x_{ij}(w)\bigr) -\pi_i\bigl(x_{ij}(w)\bigr), \]

and define the visibility score

\[ \sigma_{ij}(w)=\min_{k\ne i,j} g_{ijk}(w), \]

with the value \(+\infty\) when no competitor exists. Each competitor slack is affine in the fitted weights. Under the nondegeneracy assumption that no competitor bisector coincides with the pair bisector, \(\sigma_{ij}(w)>0\) exactly when the connector point lies in the relative interior of the unblocked pair face. A zero score marks a higher-order tie, and a negative score means that at least one competitor blocks the connector point.

For a connector-visible pair, clearance is the in-face distance from the connector point to the nearest blocking competitor hyperplane. Domain clipping can impose an additional limiting margin. Visibility is weaker than full face realization, and clearance is a geometric ranking or stability diagnostic—not a universal application-specific decision rule.

The implemented outer algorithm alternates between fitting and realization:

fit the currently active observations;
compute the resulting power diagram;
match requested boundaries and periodic images;
update the active set using hysteresis;
stop on self-consistency, a detected cycle, numerical failure, or an iteration limit.

The graph compatibility results and the convex fixed-observation estimator form the exact inner theory. The hysteretic outer loop is a practical, realization-aware algorithm with inspectable path diagnostics; it is not claimed to be globally convergent for every candidate set.

After the final active set is refitted, component-offset alignment is applied only to zero-L2 gauge components and only when exact binary64-input checks show that all component weight differences are unchanged. Positive L2 fixes the component means as part of the objective, so aligning such a fit to a previous iterate would change the optimization problem and is not performed.

Interpreting a result

A scientifically useful report should answer separate questions:

Are the observations exactly or approximately compatible?
Which relative weights are identified by the effective graph?
Which component offsets were chosen by policy or prior information?
Are hard restrictions feasible, and is there a contradiction witness if not?
Which requested pairs and periodic images are realized?
Are any endpoint cells empty?
Did outer refinement converge, cycle, fail numerically, or reach a limit?

These questions should not be collapsed into one generic “fit succeeded” flag. In particular, a solver cannot report an optimal or converged result when any reported soft-objective component or total is non-finite. A failed optional direct ADMM warm start falls back to the reference or zero initialization and does not by itself constitute solver failure.

Scope and further reading

The theory above assumes fixed sites and scalar power weights. It does not infer site positions, guarantee global convergence of the active-set loop, or claim that pairwise separator data reconstruct a complete application-specific partition.

The current calls and result objects are documented in the separator-fitting guide and API reference. Full proofs, perturbation results, synthetic experiments, periodic-image tests, and the molecular benchmark are given in the manuscript Fixed-site inverse fitting of power-diagram weights from pairwise separator data and its archived reproducibility materials.