Globally Predictable Interpolation (GPI)

• The paper’s main contribution shows that using global constraints rather than local neighborhoods yields more predictable and stable interpolation results.
• GPI is defined as a framework where intermediate states are inferred from global dependencies, with applications spanning MRI k-space, video pose spaces, and controlled text generation.
• Empirical studies reveal that integrating global structured priors with local methods or numerical schemes improves accuracy, smoothness, and semantic consistency.

Globally Predictable Interpolation (GPI) denotes a class of interpolation ideas in which intermediate states are constrained by global structure rather than inferred only from local neighborhoods or from unconstrained latent blends. In the most explicit arXiv usage represented here, GPI is formulated for accelerated MRI $k$-space interpolation as prediction from global dependencies spanning the entire $k$-space (Luo et al., 6 Aug 2025). Closely related work extends the same objective to semantically organized pose spaces for video, convex-hull interpolation of LoRA anchor models for controllable text generation, density-aware path optimization in generative models, computationally stable Gaussian-process interpolation for deterministic simulators, and transformed Chebyshev-style interpolation on equidistant grids (Shih et al., 2019, Kangaslahti et al., 2024, Rygaard et al., 30 Oct 2025, Ranjan et al., 2010, Liu et al., 22 Nov 2025). Taken together, these works suggest that interpolation becomes more predictable when the interpolating variables, objective, or numerical scheme are globally structured.

1. Conceptual scope and recurring definition

The literature does not present a single universal formalism of GPI. Instead, it presents several technically distinct formulations that share a common contrast: local predictability is treated as insufficient. In the MRI formulation, conventional GRAPPA/SPIRiT-like methods and CNN-based $k$-space methods are characterized as local because they infer missing values from nearby acquired samples or finite receptive fields, whereas GPI is defined by global dependencies across the full $k$-space (Luo et al., 6 Aug 2025). In video, “predictable” interpolation means that interpolation in latent pose space yields an image-space trajectory that is semantically understandable and geometrically regular rather than visually erratic (Shih et al., 2019). In controllable language generation, predictability means that varying interpolation weights produces smooth and consistent changes in measured output attributes (Kangaslahti et al., 2024). In generative modeling, the emphasis shifts from local smoothness to a path-level objective that favors likely transitions under a model distribution (Rygaard et al., 30 Oct 2025).

This variety is substantive rather than terminological. Some works aim at semantic controllability, some at density-aware transition paths, and some at exactness-preserving numerical interpolation. The shared theme is that interpolation is made globally organized: by explicit geometry, by convex parameterization, by a global path functional, or by a numerical procedure that stabilizes the entire predictive surface. This suggests that GPI is better understood as a research orientation than as a single algorithmic template.

A second recurring distinction concerns the object to which predictability applies. In some settings it is motion or pose; in others it is style attributes, latent-density traversal, or deterministic emulator exactness. The result is a layered notion of predictability: semantic predictability, geometric predictability, probabilistic predictability, and numerical predictability. The papers agree on the inadequacy of purely local or purely heuristic interpolation, but they operationalize the remedy in different mathematical spaces.

2. Explicit GPI in accelerated MRI

The clearest formalization appears in "Towards Globally Predictable k-Space Interpolation: A White-box Transformer Approach" (Luo et al., 6 Aug 2025). There, GPI means that missing MRI $k$-space samples are predicted from global dependencies spanning the entire $k$-space rather than from local neighborhoods. The starting point is the multi-coil measurement model

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$

followed by an annihilation-based structured low-rank regularizer

$$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$

with spectral penalty

$$\rho_\gamma(\mathbf{X}) = \mathbf{U}\, \mathrm{diag}\!\big( \ln(1+\gamma\sigma_1),\dots,\ln(1+\gamma\sigma_r) \big)\, \mathbf{V}^*.$$

The reconstruction objective combines data consistency, a global structured low-rank term, and a local SPIRiT prior:

$$\min_{\mathbf{k}} \frac{1}{2}\|M_{\Omega}\mathbf{k} - \mathbf{y}\|_2^2 + \lambda_1 R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) + \lambda_2 \|(\mathbf{G}-\mathbf{I})\mathbf{k}\|_2^2.$$

The global component is derived from annihilation rather than introduced as a generic attention block. Global annihilation filters are treated as learnable parameters, and the subgradient of the structured low-rank term induces attention:

$k$0

with multi-head aggregation

$k$1

Unlike standard black-box attention, the model uses a single matrix per head so that $k$2. This is the basis for the paper’s “white-box” designation: attention is the optimization-induced update associated with an explicit regularizer.

Unfolding gradient descent yields the stage-wise update

$k$3

where

$k$4

The implementation alternates square and linear window partitioning to reduce attention cost while preserving long-range interaction. Conceptually, the method does not abandon local predictability; it supplements it with a global prior, and the ablation results indicate that the global term alone is not sufficient.

The experiments use fastMRI knee raw data with 31 training subjects and 3 test subjects, 10 unfolded iterations, $k$5 square windows, 6 heads, Adam, and an initial learning rate of $k$6. Under random undersampling, GPI-WT reports NMSE/PSNR/SSIM of $k$7 at AF $k$8 and $k$9 at AF $k$0; under uniform masks, it reports $k$1 at AF $k$2 and $k$3 at AF $k$4, outperforming SPIRiT, KNet, Swin, DSLR, and GPI-CNN in those settings (Luo et al., 6 Aug 2025). The paper’s central claim is therefore not merely that Transformers help, but that a global, interpretable, annihilation-derived attention mechanism improves $k$5-space interpolation accuracy.

3. Structured latent-state interpolation in video and language

A closely aligned but less formal variant appears in "Video Interpolation and Prediction with Unsupervised Landmarks" (Shih et al., 2019). The paper does not use the term GPI, yet it constructs a globally organized, low-dimensional, semantically interpretable pose space. Each frame is factored into pose and appearance:

$k$6

with pose represented as $k$7 normalized activation maps and then compressed into Gaussian landmark states

$k$8

Each landmark therefore carries 2D location $k$9, covariance $k$0, and an appearance descriptor $k$1. The decoder reconstructs semantic conditioning maps from

$k$2

and combines them with appearance codes in a SPADE-style generator.

Interpolation is performed directly in the landmark parameter space. Because covariance matrices must remain positive definite, the paper interpolates the Cholesky factor $k$3 where $k$4. The state for each landmark is thus 5-dimensional in 2D, and linear interpolation is

$k$5

This is the paper’s central mechanism for approximate GPI: straight lines in a semantically grounded pose space are expected to induce understandable motion in image space. The evidence is strongest on BAIR, where interpolation is performed between the first and last frame of a sequence. The paper reports that VideoFlow’s latent interpolation is “reasonably successful” but follows a trajectory “difficult to anticipate,” whereas pose-space interpolation “move[s] predictably in a linear trajectory in image space while maintaining consistent image quality,” with SSIM comparable to SuperSlomo though slightly worse in the first and last few frames (Shih et al., 2019). The paper also states clear limitations: no formal global-topological guarantee, deterministic prediction, possible landmark identity switching, and weak modeling of backgrounds and novel appearances.

The same structured-control theme appears in "Continuous LLM Interpolation for Dynamic and Controllable Text Generation" (Kangaslahti et al., 2024). There the base model is Llama2-7b, and each style attribute is controlled by two LoRA-fine-tuned endpoint anchors. The LoRA parameterization is

$k$6

and single-attribute interpolation is written as

$k$7

Multi-attribute interpolation uses simplex weights $k$8:

$k$9

The controlled attributes are simplicity, formality, politeness, sentiment, and humor. The paper reports that varying $k$0 and $k$1 yields predictable and consistent changes in output attributes measured by RoBERTa classifiers. In the interpolation region, all attributes show smooth monotone trends; politeness, formality, and simplicity are approximately linear, while sentiment and humor show nonlinear plateaus. The paper further states that there is “surprisingly little entanglement” for most attributes, although the humor-formality-simplicity simplex and the sentiment-politeness-formality simplex show correlated regions, and extrapolation outside the convex region becomes unstable beyond roughly $k$2 and $k$3 (Kangaslahti et al., 2024). This is strong empirical evidence for approximate GPI over a multi-anchor convex hull, but it remains empirical rather than theorem-driven.

4. Path-level and density-aware interpolants in generative models

"Likely Interpolants of Generative Models" (Rygaard et al., 30 Oct 2025) shifts the locus of predictability from coordinates or weights to the entire interpolation path. Instead of defining interpolants by linear or spherical heuristics, the paper poses a global optimization problem over a discrete path $k$4 with fixed endpoints $k$5, $k$6. The relaxed objective is

$k$7

with $k$8 typically chosen as

$k$9

The first term is a discretized geodesic energy, while the second penalizes low-density regions. Predictability here means that every interior waypoint is jointly determined by a single endpoint-to-endpoint objective rather than by a local interpolation rule.

The paper also gives a control form with $k$0, derives first-order necessary conditions, and introduces the ProbGEORCE algorithm. For fixed metrics $k$1 and local derivatives

$k$2

the update is available in closed form:

$k$3

$k$4

In the Euclidean case $k$5, the method has the same complexity as gradient descent. The paper is explicit that no additional training is required; the procedure operates directly on already trained VAEs, Euclidean diffusion models, latent diffusion models, and Riemannian diffusion models.

Its main theoretical statement is local rather than absolute. Along an optimal path, the energy is approximated by

$k$6

so the effective local metric becomes

$k$7

The interpolant is therefore not globally a standard geodesic of a fixed metric, but it is locally geodesic under a density-corrected metric. Empirically, the paper reports higher-density traversal than baselines across model classes and datasets, including OASIS3, CIFAR10, CelebAHQ, and several Riemannian diffusion settings (Rygaard et al., 30 Oct 2025). In GPI terms, this is one of the strongest formulations of globally constrained interpolation, even though the optimization remains nonconvex and only local minima are guaranteed.

5. Numerical global interpolation and exactness-preserving schemes

A different but related tradition treats global predictability as numerical exactness and whole-domain stability. "A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data" (Ranjan et al., 2010) studies deterministic simulators for which the emulator should interpolate exactly rather than smooth. The obstacle is numerical near-singularity of the correlation matrix $k$8, which makes direct GP interpolation unstable. The usual nugget remedy replaces $k$9 by $\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$0, but this destroys exact interpolation at training points. The paper proposes a lower bound

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$1

with empirical recommendation $\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$2, and an iterative regularization scheme

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$3

that converges to the exact inverse:

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$4

Consequently,

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$5

The paper’s relevance to GPI is numerical rather than semantic: it formalizes a globally stable route back to the exact GP interpolator while avoiding unnecessary over-smoothing. The practical consequence is significant, since excessive smoothing can distort inferred optima in deterministic computer experiments.

"Fast and stable global interpolation based on equidistant points" (Liu et al., 22 Nov 2025) addresses another classical obstacle to predictable global interpolation: the Runge phenomenon on equidistant nodes. The proposed symmetric wave interpolation maps equidistant nodes

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$6

to Chebyshev nodes through nonlinear coordinate warps. For the first and second kinds,

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$7

$\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$8

If $\mathbf{y} = M_{\Omega}\mathbf{k} + \mathbf{n},$9 and $$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$0 are Chebyshev interpolants in the transformed variable, the final interpolants are

$$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$1

The explicit forms are

$$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$2

$$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$3

The paper gives interval-wide error bounds via transformed Chebyshev interpolation and reports that SWI suppresses Runge oscillations, often reaches target error thresholds faster than Chebyshev interpolation, and remains robust when sample values are rounded, although endpoint-near sharp features remain a weak case (Liu et al., 22 Nov 2025). Here global predictability means stable, strict global interpolation from readily available equidistant samples.

6. Limits, misconceptions, and acronym collisions

The literature imposes clear limits on what GPI should and should not mean. First, these works indicate that GPI is not synonymous with any linear interpolation. In the video setting, predictability depends on semantically stable landmark Gaussians and decoder generalization, not on linearity alone; the paper explicitly notes the lack of a formal global-topological constraint, the possibility of landmark identity switching, and the weakness of background modeling (Shih et al., 2019). In language-model interpolation, predictability is strongest inside the convex hull of LoRA anchors, weakens for correlated attributes when a non-target mixing weight exceeds roughly $$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$4–$$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$5, and becomes unstable in strong extrapolation regimes (Kangaslahti et al., 2024). In generative-model path optimization, the path is globally objective-driven but not guaranteed to be the unique global optimum, since the optimization is nonconvex and the Lagrangian relaxation may exhibit duality gaps (Rygaard et al., 30 Oct 2025).

Second, explicit GPI formulations can still depend on hybridization with local priors. The MRI paper includes the local SPIRiT term $$R\!\left(\mathbf{k};\mathbf{Q}_{[H]}\right) := \sum_{h=1}^H \mathrm{Tr}\!\left[ \rho_\gamma\!\left( (\mathbf{Q}_h\mathbf{k})^*(\mathbf{Q}_h\mathbf{k}) \right) \right],$$6 because pure global regularization may weaken local dependencies, and its practical implementation still uses square and linear windows rather than full all-to-all global attention (Luo et al., 6 Aug 2025). This makes the method a globally structured interpolation framework rather than a purely global one in the strongest possible sense.

Third, the numerical-interpolation literature shows that global predictability may mean stability or exactness rather than semantic controllability. The GP paper is restricted to deterministic computer simulators and GP emulators, and the SWI paper provides transformed-Chebyshev theory and empirical evidence but not a full floating-point stability theory, Lebesgue-constant analysis, or universal superiority guarantee (Ranjan et al., 2010, Liu et al., 22 Nov 2025). These are supporting instances of GPI-like concerns, not a single doctrine.

Finally, the acronym itself is ambiguous. In "Influence of electric fields on dielectric properties of GPI ferroelectric" (Zachek et al., 2017), GPI refers to glycinium phosphite, not interpolation. In "Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space" (Bendle et al., 2019), GPI-Space denotes a workflow management system; the paper presents a semi-numeric rational interpolation method implemented in Singular and orchestrated by GPI-Space, but it does not define “Globally Predictable Interpolation.” Any encyclopedic treatment of GPI therefore requires explicit disambiguation between the interpolation concept, the MRI-specific formalism, and unrelated acronym uses.

In aggregate, GPI is best characterized as an emerging research idea that seeks interpolation spaces, objectives, or numerical schemes in which global structure renders intermediate states controllable, stable, or density-respecting. Its explicit mathematical realization is strongest in accelerated MRI, its semantic approximations are clearest in structured latent video and language interpolation, its path-level generalization is most rigorous in density-aware generative-model interpolation, and its numerical analogue appears in exactness-preserving GP and transformed-node global interpolation. The concept is coherent across these settings, but it is not yet unified by a single formal theory.