Local Dimension: Invariants & Applications

• Local Dimension is a family of invariants that measures complexity locally, capturing per-element frequencies in posets, scaling exponents in measures, and recurrence in dynamics.
• It enables the identification of structural inhomogeneities in networks and estimates tangent-space dimensions in deep representations to optimize data analysis.
• Arithmetic and multifractal theories apply local dimension to track minimal parameterizations and local scaling behaviors, bridging concepts in geometry, dynamics, and algebra.

Searching arXiv for recent and foundational papers on “local dimension” across the main usages represented in the provided material. Local dimension denotes a family of invariants that quantify complexity at a point, element, state, or scale rather than only globally. In the literature represented here, the term is used for a frequency parameter attached to local realizers of posets, for the pointwise scaling exponent of measures, for an attractor-dimension inferred from recurrence statistics, for node- and scale-dependent descriptors of networks, for local intrinsic or tangent-space dimension in data analysis and deep representations, and for the minimal number of parameters needed to realize local Galois extensions over number fields (Kim et al., 2018, Yuan, 2018, Buschow et al., 2018, Peach et al., 2021, Zhang et al., 2017, König et al., 2020).

1. Posets and order-theoretic local dimension

For a finite poset $P=(P,\le)$, a partial linear extension is a linear extension of some induced subposet. A local realizer is a nonempty family $\mathcal L$ of partial linear extensions such that every comparable pair $x<y$ appears in the correct order in some $L\in\mathcal L$, while every incomparable pair $x\parallel y$ is witnessed in both directions across the family. If

$$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$

then

$$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$

and one immediately has $\operatorname{ldim}(P)\le \dim(P)$ (Kim et al., 2018).

This invariant separates sharply from classical Dushnik–Miller dimension. The standard example has dimension $n$ but local dimension $3$, so the two parameters can be arbitrarily far apart (Kim et al., 2018). More generally, the maximum local dimension of an $\mathcal L$0-element poset is $\mathcal L$1, rather than Hiraguchi’s $\mathcal L$2 bound for ordinary dimension. The upper bound is obtained by passing to a height-two split poset and covering the incomparability graph by complete bipartite subgraphs via the Erdős–Pyber theorem; difference graphs, whose vertices in one bipartition class have nested neighborhoods, provide the combinatorial bridge between such graph covers and families of partial linear extensions (Kim et al., 2018).

Boolean lattices supplied one of the central test cases. For the $\mathcal L$3-dimensional Boolean lattice $\mathcal L$4, the 2018 bounds were

$\mathcal L$5

hence $\mathcal L$6 (Kim et al., 2018). A later result showed that for every $\mathcal L$7, the local dimension of the Boolean lattice $\mathcal L$8 is strictly less than $\mathcal L$9, with explicit upper bounds

$x<y$0

from two different constructions (Hodor et al., 11 Dec 2025).

Structural graph-theoretic behavior is markedly different from that of Boolean dimension. Local dimension is unbounded for planar posets, is not bounded by the maximum local dimension of the blocks, and is not bounded in terms of the tree-width of the cover graph alone, even when tree-width is at most $x<y$1 (Bosek et al., 2017). By contrast, it is bounded in terms of the path-width of the cover graph (Barrera-Cruz et al., 2017). The removable-pair program gives a different line of structure theory: the removable-pair conjecture asks whether every poset $x<y$2 with $x<y$3 has $x<y$4 satisfying $x<y$5; this was proved for height-two posets, and a removable-quadruple theorem shows that every $x<y$6 poset has four elements whose deletion increases local dimension by at most $x<y$7 (Kim et al., 2018).

2. Pointwise local dimension of measures and multifractal theory

In geometric measure theory, local dimension is the pointwise scaling exponent of a measure. For a Borel measure $x<y$8 on a metric space and $x<y$9,

$L\in\mathcal L$0

when the limit exists, with lower and upper local dimensions defined by $L\in\mathcal L$1 and $L\in\mathcal L$2 respectively (Yuan, 2018). This quantity records the local power-law behavior of $L\in\mathcal L$3 as $L\in\mathcal L$4, and exact dimensionality means that the lower and upper local dimensions coincide $L\in\mathcal L$5-almost everywhere (Käenmäki et al., 2015).

For Moran constructions in complete doubling metric spaces, local dimensions admit cylinder-based descriptions. If $L\in\mathcal L$6 is a Moran construction and $L\in\mathcal L$7 is coded by $L\in\mathcal L$8, then

$L\in\mathcal L$9

Under additional nondegeneracy and $x\parallel y$0-summability assumptions, the local dimension exists for $x\parallel y$1-almost every point and is given by the ratio of average entropy to average log-contraction (Käenmäki et al., 2015).

Existence is not automatic. A class of Moran measures was constructed for which the local dimension fails to exist at every point of the support: lower and upper local dimensions are distinct everywhere, and the classical $x\parallel y$2-spectrum $x\parallel y$3 does not exist. Nevertheless, full Hausdorff and packing spectra can still be obtained for the level sets defined by lower and upper local dimensions, yielding a generalized multifractal formalism (Yuan, 2018). This is one of the clearest demonstrations that pointwise local dimension and global free-energy functions can decouple.

For self-affine measures, several exact-dimensionality results were established. In a generic setting of self-affine and almost self-affine constructions, projected Bernoulli or Gibbs-type measures are exact dimensional, with local dimension equal almost everywhere to the information dimension and characterized by the zero of a superadditive pressure functional (Falconer et al., 2011). For invariant measures on infinitely generated self-affine sets, the local dimension exists for almost all translations, equals the minimum of the local Lyapunov dimension and the ambient dimension, and implies exact dimensionality for ergodic measures (Rossi, 2013). In one-dimensional self-similar settings satisfying the finite neighbour condition, the set of attainable local dimensions is a finite union of compact intervals, with the number of intervals bounded above by the number of non-trivial maximal strongly connected components in a finite directed graph construction (Hare et al., 2021). For dominated irreducible planar self-affine measures, the local dimension spectrum can be analyzed by a symbolic Legendre-transform formalism, and deterministic multifractal results were obtained in both the $x\parallel y$4 and $x\parallel y$5 dimension ranges under separation and projection hypotheses (Batsis et al., 2024).

The random $x\parallel y$6-transformation provides a further explicit model. For generalised multinacci $x\parallel y$7, the local dimensions of the natural invariant measure $x\parallel y$8 are bounded in terms of the frequency $x\parallel y$9 with which the orbit visits switch intervals, and when $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$0 exists the exact local dimension is

$$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$1

Points of unique $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$2-expansion satisfy $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$3 and therefore realize a distinct local-dimension value (Dajani et al., 2012).

A recent extension concerns convolution. For regular measures $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$4 on a metric group with translation-invariant metric, upper and lower local-dimension bounds for $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$5 can be transferred to $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$6 on interior points of $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$7. On the real line, under a “no big gaps” assumption on $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$8, one obtains corresponding interior-point bounds for $$p(x,\mathcal L)=|\{L\in\mathcal L:x\text{ appears in }L\}| \quad\text{and}\quad \mu(\mathcal L)=\max_{x\in P} p(x,\mathcal L),$$9; at special boundary or gap-endpoint configurations, exact formulas reduce the local dimension of the convolution to that of an associated product measure (Hare et al., 25 Feb 2025).

3. Local attractor dimension in dynamical systems and climate simulations

In dynamical systems, local dimension is attached to an invariant attractor. For a measure-preserving system $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$0 and a point $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$1 on the attractor, one assumes

$$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$2

and defines

$$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$3

This local attractor dimension is linked to recurrence statistics, predictability of individual configurations, and the information gained from observing them (Buschow et al., 2018).

Extreme-value theory yields a practical estimator. For the distance observable $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$4, the conditional exceedance distribution is a Generalized Pareto law with shape $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$5 and scale $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$6. If $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$7 are distances to a reference state $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$8, $$\operatorname{ldim}(P)=\min\{\mu(\mathcal L):\mathcal L\text{ is a local realizer}\},$$9 is the distance threshold corresponding to a high quantile $\operatorname{ldim}(P)\le \dim(P)$0, and $\operatorname{ldim}(P)\le \dim(P)$1 exceedances satisfy $\operatorname{ldim}(P)\le \dim(P)$2, then the maximum-likelihood estimator is

$\operatorname{ldim}(P)\le \dim(P)$3

with asymptotic standard error

$\operatorname{ldim}(P)\le \dim(P)$4

where $\operatorname{ldim}(P)\le \dim(P)$5 is the threshold quantile (Buschow et al., 2018).

The implementation is recurrence-based. One fixes a reference time, computes distances from the reference state to the rest of the orbit, optionally excludes immediate temporal neighbors to avoid short-record bias, applies the observable $\operatorname{ldim}(P)\le \dim(P)$6, and estimates $\operatorname{ldim}(P)\le \dim(P)$7 from the exceedances above a high threshold. The paper recommends high thresholds $\operatorname{ldim}(P)\le \dim(P)$8, enough exceedances $\operatorname{ldim}(P)\le \dim(P)$9–$n$0, and an explicit likelihood-ratio test comparing the theoretically required $n$1 model with a free-$n$2 alternative (Buschow et al., 2018).

In the PUMA climate-model experiments, using 1000 years of daily sea-level-pressure data over the North Atlantic, the mean estimated local dimension rose from approximately $n$3 to approximately $n$4 as the threshold increased from $n$5 to $n$6, while remaining far below the ambient dimension $n$7. Likelihood-ratio rejections of $n$8 dropped from about $n$9 at $3$0 to about $3$1 at $3$2, and the estimated shape parameter moved toward $3$3 from about $3$4 to about $3$5. Even with a 1000-year record there was no convergence to the reported global information-dimension proxy $3$6, indicating slow convergence; in 5-year segments, retaining direct temporal neighbors biased $3$7 downward by up to $3$8 at high thresholds (Buschow et al., 2018).

The same study used local dimension to complement principal component analysis. Days below the $3$9 local-dimension quantile exhibited a pronounced Atlantic dipole, described as NAO$\mathcal L$00-like, and showed enhanced zonal wavenumber $\mathcal L$01 and $\mathcal L$02 power; days above the $\mathcal L$03 quantile were nearly climatological. The paper interprets high $\mathcal L$04 as many degrees of freedom being active locally and therefore lower instantaneous predictability, while low $\mathcal L$05 indicates a more effective low-dimensional regime (Buschow et al., 2018).

4. Local dimension on networks

One network-theoretic definition is based on metric balls in graph distance. For a connected unweighted network, let $\mathcal L$06 be the number of nodes at exactly distance $\mathcal L$07 from node $\mathcal L$08, and

$\mathcal L$09

the number of nodes within distance at most $\mathcal L$10. If $\mathcal L$11, then the scale-dependent local dimension is defined by

$\mathcal L$12

and in discrete form

$\mathcal L$13

This construction is intended to detect regions with distinct dimensional structure and to identify borders (Silva et al., 2012).

Applied to power-grid networks, the resulting dimension profiles separated two geographically embedded systems. For the continental European power grid $\mathcal L$14, the maximum average local dimension was reported as $\mathcal L$15; for the western-US grid $\mathcal L$16, it was $\mathcal L$17. The European grid was therefore described as substantially more planar, whereas the western-US system had topological dimension higher than its intrinsic embedding-space dimension $\mathcal L$18 (Silva et al., 2012).

A second network definition is diffusion-based and intrinsic. In Euclidean space, diffusion from a point source has a Green’s function whose peak time and amplitude determine the dimension; on graphs, one replaces the Laplacian by the normalized graph Laplacian $\mathcal L$19, runs consensus dynamics

$\mathcal L$20

and for each ordered pair $\mathcal L$21 extracts the peak $\mathcal L$22 of $\mathcal L$23. The pairwise relative dimension is then

$\mathcal L$24

and the local dimension at scale $\mathcal L$25 is the average

$\mathcal L$26

As $\mathcal L$27, this approaches the stationary local dimension (Peach et al., 2021).

This diffusion-based formalism was validated on one-dimensional lines and two-dimensional grids, where the peak of the global dimension $\mathcal L$28 converged to the Euclidean dimension as $\mathcal L$29, and on Delaunay triangulations with inhomogeneous “added mass,” where local dimension reflected boundary effects and diffusion-lens behavior (Peach et al., 2021). In applications, local dimension correlated with residue flexibility in structural protein graphs, with Pearson $\mathcal L$30 across 12 proteins at mid-to-long timescales, while the global dimension correlated with average protein rigidity at $\mathcal L$31. In epidemic simulations, the optimal predicting scale $\mathcal L$32 for infectivity grew roughly linearly with transmission probability $\mathcal L$33 below the threshold and diverged near $\mathcal L$34, where the correlation between $\mathcal L$35 and infectivity approached unity. For graph classification, a Random Forest using the mean, standard deviation, and skewness of stationary $\mathcal L$36 distributions achieved $\mathcal L$37 accuracy on Erdős–Rényi, Watts–Strogatz, and Barabási–Albert ensembles (Peach et al., 2021).

5. Local intrinsic dimension in data analysis and deep representations

In one machine-learning usage, local dimension is the tangent-space dimension of a representation manifold. If $\mathcal L$38 is the set of activations produced by a deep network at a given layer and $\mathcal L$39, then the local dimension at $\mathcal L$40 is

$\mathcal L$41

A practical estimator constructs a cloud of nearby activations around an anchor image, forms the activation matrix $\mathcal L$42, computes its singular values, and identifies a dramatic drop. The paper uses the rule

$\mathcal L$43

with a very large threshold, such as $\mathcal L$44, and sets the estimated local dimension to $\mathcal L$45. Three perturbation schemes were examined—random cropping, small rotations, and additive Gaussian noise—and Gaussian noise was reported as the closest approximation to an arbitrary infinitesimal perturbation in image space (Zhang et al., 2017).

For VGG-19, the reported local dimensions declined rapidly with depth and were similar across the ImageNet classes Persian Cat, Container Ship, and Volcano. In the Conv5 block, the estimates were approximately $\mathcal L$46, $\mathcal L$47, $\mathcal L$48, and $\mathcal L$49 for Conv5_1 through Conv5_4; at pool5, fc6, fc7, and fc8 they were approximately $\mathcal L$50, $\mathcal L$51, $\mathcal L$52, and $\mathcal L$53, respectively (Zhang et al., 2017). The paper interprets these measurements as evidence that deeper layers compress nuisance directions and increasingly constrain the representation to lower-dimensional tangent spaces.

A different data-analytic notion is local intrinsic dimension, defined through nearest-neighbor statistics. If $\mathcal L$54 are the first and second nearest-neighbor distances from $\mathcal L$55, then

$\mathcal L$56

follows the Pareto density

$\mathcal L$57

under the local-uniformity assumptions used in the TWO-NN framework. The local intrinsic dimension of $\mathcal L$58 is then the intrinsic dimension $\mathcal L$59 of the manifold to which the point is assigned (Allegra et al., 2019).

The Hidalgo algorithm casts this as a Bayesian mixture-of-Pareto problem with neighborhood homogeneity. It infers latent labels $\mathcal L$60, manifold-specific dimensions $\mathcal L$61, and mixing weights $\mathcal L$62, using the likelihood

$\mathcal L$63

together with a soft constraint that $\mathcal L$64-nearest neighbors tend to share the same label. The defaults reported were $\mathcal L$65 and $\mathcal L$66, and the practical complexity summary was $\mathcal L$67 for moderate $\mathcal L$68 (Allegra et al., 2019). In the reported applications, Hidalgo found $\mathcal L$69 manifolds with dimensions approximately $\mathcal L$70 in a protein-folding trajectory, $\mathcal L$71 manifolds with dimensions approximately $\mathcal L$72 in fMRI voxel time series, and $\mathcal L$73 manifolds with dimensions approximately $\mathcal L$74 in financial firm data (Allegra et al., 2019).

6. Arithmetic local dimension and terminological relationships

In arithmetic geometry, local dimension has a different meaning. Let $\mathcal L$75 be a finite group and $\mathcal L$76 a number field. The local dimension $\mathcal L$77 is the smallest integer $\mathcal L$78 for which there exists a finite family of $\mathcal L$79-extensions

$\mathcal L$80

such that for all but finitely many finite places $\mathcal L$81 of $\mathcal L$82, every $\mathcal L$83-extension of $\mathcal L$84 is a specialization of some $\mathcal L$85 at a $\mathcal L$86-rational place of $\mathcal L$87 (König et al., 2020). The Hilbert–Grunwald dimension $\mathcal L$88 is defined similarly using simultaneous local conditions, and the inequalities

$\mathcal L$89

hold by definition (König et al., 2020).

The main theorem gives a uniform upper bound: $\mathcal L$90 More precisely, for every finite group $\mathcal L$91 and number field $\mathcal L$92, there exists a single $\mathcal L$93-extension $\mathcal L$94 with $\mathcal L$95 such that, away from finitely many primes, every local $\mathcal L$96-extension of $\mathcal L$97 is realized by specialization. If $\mathcal L$98 admits a generic extension over $\mathcal L$99, then one may take $x<y$00 and obtain $x<y$01 as well (König et al., 2020). The same paper records a lower bound: if $x<y$02 contains any non-cyclic abelian subgroup, then $x<y$03; for cyclic groups or certain semidirect products of cyclic groups of coprime prime-power orders, $x<y$04 is described as achievable in some cases (König et al., 2020).

Taken together, these usages suggest that “local dimension” is a shared label for several distinct localization procedures rather than a single formal invariant. In posets it counts the maximum per-element frequency in a local realizer; in measure theory it is a pointwise scaling exponent of $x<y$05; in dynamical systems it is recovered from recurrence extremes; in networks it is extracted from ball growth or diffusion transients; in machine learning it is a tangent-space or manifold-membership quantity; and in arithmetic geometry it is the minimal transcendence degree required to parametrize almost all local realizations (Kim et al., 2018, Yuan, 2018, Buschow et al., 2018, Peach et al., 2021, Zhang et al., 2017, König et al., 2020). A plausible implication is that the enduring value of the term lies in replacing a global notion of dimension by one conditioned on a local neighborhood, local witness system, or local field.