Centralizing Bases in Diverse Domains

• Centralizing bases is a set of strategies that replace local, unstable structures with shared, global reference points across diverse application domains.
• In zero-shot sketch-based image retrieval, ACNet employs learnable proxies as centralizing bases, leading to more compact class clusters and improved retrieval performance.
• Centralizing bases also optimize wireless processing, k-center facility placement, and algebraic constructions by aligning decentralized units into consistent, globally defined frameworks.

“Centralizing bases” is not a single invariant technical term. In contemporary research usage it denotes a family of centralization operations that introduce shared anchors, shared processors, or shared canonical objects in order to reduce variance, fragmentation, or ambiguity. In zero-shot sketch-based image retrieval, it refers to learnable proxy centers in an embedding space; in wireless systems, to centralizing baseband processing or pooling service capacity; in operations research, to the siting of physical bases under a $k$-center objective; in classical shadows, to measurement bases that make the measurement channel central on visible isotypic components; in cluster and skein algebras, to proving that independently constructed bases coincide; and in whole-cell modeling, to assembling a central warehouse of heterogeneous biological data (Ren et al., 2021, Alegría et al., 10 Jan 2025, West et al., 1 Apr 2026, Mandel et al., 2023, Chew et al., 2021).

1. Terminological range and recurrent structure

Across these literatures, the word “bases” names different mathematical or physical objects. In ACNet, the “bases” are learnable class proxies $p_c$ in the embedding space, not seen categories themselves and not inference-time prototypes. In decentralized multi-antenna architectures and C-RAN, “bases” refers to base stations, radio access points, or baseband resources. In classical shadows, the relevant bases are measurement bases adapted to the representation-theoretic structure of a unitary ensemble. In cluster and skein algebra, the issue is the relation among topological, canonical, and quantized bases. In whole-cell modeling, the emphasis shifts from basis objects to the centralized “base” of data needed for model construction (Ren et al., 2021, Alegría et al., 10 Jan 2025, West et al., 1 Apr 2026, Mandel et al., 2023, Chew et al., 2021).

A recurrent pattern nevertheless appears. Centralization introduces a common reference structure that replaces unstable pairwise relations or scattered local decisions with classwise aggregation, global processing, or canonical decomposition. In representation learning this means proxy-centered losses; in wireless architectures it means CPU- or BBU-centered computation and pooled complexity; in classical shadows it means block-scalar measurement channels; in cluster theory it means collapsing distinct natural bases into one canonical basis. A common misconception is therefore to assume that “centralizing bases” has a uniform formal definition across fields. The literature instead supports a domain-specific interpretation whose technical meaning depends on what is being centralized.

2. Proxy-centered embedding learning in ACNet

In ACNet for zero-shot sketch-based image retrieval, centralizing bases are the learnable proxies assigned to seen classes during training. The feature extractor $\Phi$ is a ResNet-50 backbone that produces an embedding $x \in \mathbb{R}^d$, typically with $d=512$, followed by $L_2$ normalization. Each seen class $c$ has a learnable proxy $p_c \in \mathbb{R}^d$, and the set of all proxies is denoted $Z$. These proxies act as class centers in the embedding space, so that sketch features, synthesized-photo features, and real-photo features of class $y$ are pulled toward $p_c$0 and pushed away from the other proxies. ACNet uses the proxy-based NormSoftmax loss with temperature $p_c$1, equivalently a scaled cosine classifier with $p_c$2:

$p_c$3

The loss is evaluated on three streams—real sketches $p_c$4, real photos $p_c$5, and synthesized images $p_c$6 via the chainer loss $p_c$7—and summed as

$p_c$8

The full objective is

$p_c$9

with $\Phi$0 and $\Phi$1 in experiments (Ren et al., 2021).

The centralizing effect is explicit. Because gradients are aggregated per proxy from all samples of a class, training is less sensitive to the high-variance behavior of pair or triplet mining, and compact classwise clusters emerge around the proxies. This is especially important because ACNet is jointly trained with a lightweight forward-only CycleGAN-style sketch-to-photo generator $\Phi$2 and a PatchGAN discriminator $\Phi$3. Retrieval guidance is supplied by backpropagating the chainer loss through $\Phi$4 into $\Phi$5, so the generator is encouraged to produce synthesized images whose embeddings fall near the correct proxy and away from the others. ACNet does not use angular-margin variants such as CosFace or ArcFace, favoring the vanilla normalized softmax because its centralizing effect stabilizes joint optimization in the presence of noisy synthesized inputs (Ren et al., 2021).

The empirical evidence for centralization is substantial. On Sketchy Extended with ResNet-50 and 512-dimensional embeddings, triplet loss yields $\Phi$6, whereas NormSoftmax yields $\Phi$7. On TU-Berlin Extended, a progression from the baseline $\Phi$8 to the full joint model shows monotonic gains: baseline $\Phi$9 reaches $x \in \mathbb{R}^d$0; adding $x \in \mathbb{R}^d$1 gives $x \in \mathbb{R}^d$2; adding $x \in \mathbb{R}^d$3 gives $x \in \mathbb{R}^d$4; adding the $x \in \mathbb{R}^d$5 term gives $x \in \mathbb{R}^d$6; and full ACNet reaches $x \in \mathbb{R}^d$7 and $x \in \mathbb{R}^d$8. The same work reports reduced sketch-photo domain distances after synthesis, more centralized clusters in t-SNE visualizations, and state-of-the-art performance on TU-Berlin Extended and strong results on Sketchy Extended, including $x \in \mathbb{R}^d$9 up to $d=512$0 with ResNet-50 and 512-dimensional embeddings (Ren et al., 2021).

The zero-shot aspect is decisive for interpretation. Proxies exist only for seen classes and are not used at test time for retrieval on unseen classes. Generalization instead arises from the category-agnostic and domain-agnostic structure induced during training. This suggests that, in ACNet, “centralizing bases” is best understood as a training-time geometric regularization mechanism rather than a test-time prototype system.

3. Centralized processing, base stations, and pooled capacity

In wireless systems, centralizing bases usually means moving signal processing or service capacity away from distributed local units and into a central processor. A narrowband uplink WAX architecture models decentralized preprocessing as

$d=512$1

with $d=512$2, block-diagonal local filters $d=512$3, a fixed combiner $d=512$4 that reduces the dimension from $d=512$5 antenna streams to $d=512$6 forwarded streams, and CPU-side linear processing $d=512$7. Under unitary constraints, each local block satisfies $d=512$8 and $d=512$9, so $L_2$0 is semi-unitary and the effective noise remains white. The unconstrained WAX framework is information-lossless when

$L_2$1

and the unitary-constrained variant characterizes all information-lossless semi-unitary transforms by

$L_2$2

with $L_2$3 and $L_2$4. When exact realization by $L_2$5 is impossible, the proposed alternating SVD procedure maximizes

$L_2$6

equivalently minimizing $L_2$7. In the reported setup $L_2$8, $L_2$9, and $c$0, the proposed unitary algorithm outperforms both polar projection of the unconstrained solution and random isotropic unitary choices; information-lossless operation is achieved for sufficiently large $c$1, including $c$2 for $c$3 and $c$4 for $c$5 (Alegría et al., 10 Jan 2025, Liu et al., 2015, Rost et al., 2015, Xu, 2012).

The same centralization logic appears in C-RAN. “Baseband-up centralization” places BBUs in a central office or data center while RRHs perform radio digitization at remote sites. A graph-based model writes the baseband chain as a directed graph $c$6 whose nodes are atomic functions such as FFT/IFFT, coding/decoding, modulation/demodulation, and MIMO transmission or reception, while edge weights represent information flows. Function placement becomes a clustering problem $c$7 with computational cost

$c$8

and fronthaul cost

$c$9

subject to path-delay constraints $p_c \in \mathbb{R}^d$0. The reported genetic algorithm uses rolling-wheel selection, dispersive crossover, graph-based mutation with probability $p_c \in \mathbb{R}^d$1, population size $p_c \in \mathbb{R}^d$2, and a delay penalty factor $p_c \in \mathbb{R}^d$3. The resulting design guidance is consistent: decoding is always centralized because of its high complexity and delay impact; CoMP-related MIMO functions are centralized when mutual edges would otherwise create expensive inter-site traffic; and redundancy-inflating operations such as modulation and precoding may be moved toward RRHs when fronthaul is the bottleneck (Alegría et al., 10 Jan 2025, Liu et al., 2015, Rost et al., 2015, Xu, 2012).

Centralization also affects computational provisioning. For centralized RAN, the pool-level computational outage probability is defined by

$p_c \in \mathbb{R}^d$4

and the complexity-rate tradeoff measures marginal achievable rate per marginal compute. The reported results show strong pooling gains: computational diversity at $p_c \in \mathbb{R}^d$5 is approximately $p_c \in \mathbb{R}^d$6 for $p_c \in \mathbb{R}^d$7 dB, $p_c \in \mathbb{R}^d$8 for $p_c \in \mathbb{R}^d$9 dB, and $Z$0 for $Z$1 dB; at $Z$2 RAPs, the complexity-rate tradeoff is approximately $Z$3, $Z$4, and $Z$5 for those same margins, with higher asymptotic values $Z$6, $Z$7, and $Z$8, respectively. A queueing-theoretic analog reaches the same qualitative conclusion: when a fraction $Z$9 of service capacity is centralized and assigned by longest-queue-first, the heavy-traffic mean queue length scales as

$y$0

whereas the fully local case $y$1 scales as $y$2. This suggests that even a thin centralized layer can suppress the long-tail behavior that dominates decentralized systems under stress (Alegría et al., 10 Jan 2025, Liu et al., 2015, Rost et al., 2015, Xu, 2012).

4. Geographical placement of physical bases

A different usage concerns the literal siting of bases, depots, or hubs. In the $y$3-center framework, one chooses $y$4 locations to minimize the worst-case distance from any demand point to its nearest base:

$y$5

The paper on geographical placement studies complete undirected graphs with metric distances satisfying the triangle inequality, distinguishes free placement, infrastructural placement, and node placement, and compares several algorithms, including farthest-first $y$6-approximation, $y$7-means variants, GRASP, evolutionary methods, and the proposed Dragoon heuristic (Hillmann et al., 2020).

Dragoon begins from an “orientation node” approximating the one-center optimum, then seeds the $y$8 centers by the farthest-first $y$9-Approx strategy. It refines the solution iteratively by reassigning demand vertices to their nearest center, testing candidate replacements for each center while holding the others fixed, and accepting only global improvements in the maximum distance

$p_c$00

When $p_c$01 is unchanged, tie-breaking prefers a lower mean distance

$p_c$02

For free placement, candidate points are searched on a grid with spacing $p_c$03, and if no improvement is found the algorithm bisects the granularity,

$p_c$04

until the maximal accepted deviation is reached. For node placement, candidates are restricted to demand points in the current cluster (Hillmann et al., 2020).

The reported experiments use more than ten scenarios with $p_c$05 to $p_c$06 demand vertices, Euclidean distances in $p_c$07D, and performance measures including maximum distance, $p_c$08 quantile, median, and average distance. Several quantitative regularities are emphasized. Free placement performs better than node placement, with an average distance deviation of approximately $p_c$09 and a worst case of up to $p_c$10; node placement typically needs about two additional centers to match free-placement performance, with a worst case of up to six additional centers. Dragoon consistently improves on the farthest-first baseline: at $p_c$11, the maximum normalized distance is $p_c$12 for $p_c$13-Approx, $p_c$14 for Dragoon with node placement, and $p_c$15 for Dragoon with free placement; at $p_c$16, the corresponding values are $p_c$17, $p_c$18, and $p_c$19; at $p_c$20, they are $p_c$21, $p_c$22, and $p_c$23. The study also reports a diminishing-return threshold: placing about $p_c$24 of demand points as centers yields the region beyond which an extra center improves the maximum distance by less than $p_c$25 on average (Hillmann et al., 2020).

In this setting, “centralizing bases” does not mean collapsing processing into a single location. It means solving a min–max coverage problem for a finite number of strategically chosen bases, typically under geometric or infrastructural constraints.

5. Centralizing measurement bases in classical shadows

In the theory of classical shadows with arbitrary group representations, centralizing bases are measurement bases that make the measurement channel central on each visible isotypic component of operator space. Let $p_c$26 be a compact group, $p_c$27 a finite-dimensional Hilbert space carrying a unitary representation $p_c$28, and $p_c$29 equipped with the adjoint action. A basis $p_c$30 is a Fourier basis if it is adapted to the irreducible decomposition of $p_c$31. It is $p_c$32-centralizing when the measurement channel satisfies

$p_c$33

and it is centralizing if this holds for every visible $p_c$34, so that

$p_c$35

Equivalently, a centralizing basis makes $p_c$36 central in the commutant of the $p_c$37-action on $p_c$38 (West et al., 1 Apr 2026).

The paper’s main constructive mechanism is the non-degenerate commuting-subgroup eigenbasis. If $p_c$39 is abelian and the restriction of each $p_c$40-irrep to $p_c$41 is multiplicity-free, then a simultaneous eigenbasis of $p_c$42 is centralizing, and the channel becomes

$p_c$43

where $p_c$44 and $p_c$45. The inversion is therefore exact and scalar:

$p_c$46

This collapses a potentially large dense inversion problem to blockwise rescaling on visible isotypic components (West et al., 1 Apr 2026).

The consequences propagate to estimation and sample complexity. For an observable $p_c$47, the variance of the single-shot estimator obeys

$p_c$48

with stronger $p_c$49-type bounds under positivity or tight-frame hypotheses. Visibility is explicit: if $p_c$50, the corresponding component is invisible under the chosen basis. The framework unifies earlier multiplicity-free analyses and extends them to settings with nontrivial multiplicities, while keeping the post-processing analytic rather than numerical (West et al., 1 Apr 2026).

The examples show how the representation theory determines performance. For the spin-$p_c$51 irrep of $p_c$52 in the weight basis of the maximal torus, the visible decomposition is $p_c$53 and each spin-$p_c$54 block has $p_c$55; the protocol is tomographically complete. For the tensor representation $p_c$56 on $p_c$57 qubits in the Schur basis, permutation-invariant operators lie in the visible space and admit polynomial sample complexity. For the symmetric group $p_c$58 in its permutation representation, some blocks are invisible, including the standard irrep with $p_c$59. For $p_c$60, the visibility profile depends strongly on whether one measures in a weight basis or in the computational basis of the real-shadows protocol. For the exceptional group $p_c$61, the $p_c$62-dimensional fundamental irrep yields visible blocks with $p_c$63, $p_c$64, $p_c$65, and $p_c$66 (West et al., 1 Apr 2026).

6. Canonical-basis centralization in cluster and skein algebras

In cluster and skein algebra, centralizing bases means proving that several independently defined bases are in fact the same basis. The paper on bracelets and theta bases establishes that the topological bracelets bases of Fock–Goncharov and Musiker–Schiffler–Williams coincide with the theta bases of Gross–Hacking–Keel–Kontsevich, together with their quantum versions, and that the Fock–Goncharov canonical coordinates on cluster Poisson varieties parameterizing framed $p_c$67-local systems also coincide with the associated theta functions. For unpunctured surfaces, the quantum bracelets form the atomic basis of the quantum skein algebra $p_c$68; more generally, the identifications hold with punctures and arbitrary coefficients, with a caveat for notched arcs in once-punctured tori (Mandel et al., 2023).

The basis elements themselves have explicit algebraic descriptions. For a weighted simple multicurve $p_c$69, the bracelet element is

$p_c$70

where arcs contribute ordinary powers and a simple loop $p_c$71 contributes $p_c$72 through the Chebyshev polynomials of the first kind,

$p_c$73

The paper proves that these bracelet elements equal theta functions indexed by their $p_c$74-vectors. In the coefficient-free classical setting, each notched arc in a once-punctured closed torus component contributes an extra factor $p_c$75, so that $p_c$76 in that exceptional case (Mandel et al., 2023).

The proof strategy centralizes several technical constructions. Cutting and gluing of surfaces correspond to gluing frozen indices in seeds and then unfreezing, while preserving theta functions. Annular loops are reduced to the twice-marked annulus and analyzed by explicit broken-line computations in the Kronecker scattering diagram. Non-annular loops are handled by universal positivity, atomicity of the theta basis, and Dehn twist arguments. For once-punctured closed surfaces, covering and folding constructions are used, and in genus $p_c$77 an extra wall in the scattering diagram explains the exceptional scalar factor (Mandel et al., 2023).

The conceptual consequence is strong. Topological bases, scattering-diagram bases, and canonical coordinate bases are no longer parallel constructions but a single basis viewed through different formalisms. This implies strong positivity and atomicity as corollaries: bracelet bases are strongly positive and atomic because theta bases are.

7. Further extensions: explicit centralizer-code bases and centralized data foundations

A different algebraic extension appears in twisted centralizer codes. For $p_c$78 and $p_c$79, the twisted centralizer space is

$p_c$80

For $p_c$81, the key structural identity is

$p_c$82

which yields an exact dimension formula in terms of the elementary divisors of $p_c$83:

$p_c$84

The paper also gives a constructive basis algorithm: compute the primary cyclic decomposition of $p_c$85, transport it to $p_c$86 through twisted polynomials, identify matching primary components, and assemble bases of the corresponding hom spaces from seed homomorphisms and their iterates under $p_c$87. In the special case $p_c$88, the dimension is simply $p_c$89, with a basis obtained by placing basis vectors of $p_c$90 into individual columns (Muchlis et al., 2021, Chew et al., 2021).

A broader infrastructural usage moves from basis vectors to data foundations. Whole-cell modeling requires genome annotation, transcriptomics, proteomics, metabolomics, kinetic parameters, interaction networks, imaging, and phenotype assays, but these data are scattered across repositories and articles, described with inconsistent formats, identifiers, units, and metadata. The proposed remedy is a central data warehouse with raw/ingest, curated/staging, and harmonized/production layers; multi-modal storage spanning object stores, relational databases, and graph or RDF systems; and standardization through formats and ontologies such as ISA-Tab, MultiCellDS, BioPAX, SBTab, ObjTables, BpForms, BcForms, and RightField. The paper identifies Datanator as a prototypical metadatabase that normalizes key data and supports “clouds of measurements” around related molecules and interactions. Here centralization is not a geometric or algebraic operation but an organizational precondition for reproducible model construction, calibration, and cross-evaluation (Muchlis et al., 2021, Chew et al., 2021).

Taken together, these extensions reinforce the article’s central point: “centralizing bases” is best treated as a family of structurally related strategies. Whether the objects are class proxies, baseband processors, service hubs, measurement bases, canonical algebraic bases, centralizer-code bases, or multi-omic datasets, centralization operates by replacing fragmented local structure with shared global reference structure.