Dimensionality Reduction Equals Flow Redistribution: The Engineering Essence of Point Set Planarization

I. Engineering Essence of Dimensionality Reduction Transformation

1.1 Engineering Mechanism of Topological Evolution in Dimensionality Reduction

Within the geometric framework of Multi-Origin Curvature (MOC), the continuous dimensional transition from three-dimensional fractals to two-dimensional fractals is essentially an asymptotic planarization of spatial point sets accompanied by global structured flow redistribution. This evolutionary mechanism differs fundamentally from conventional geometric projection methods featuring truncation, elimination and quality degradation: the transformation never deletes topological nodes or severs branch paths. The system adaptively adds local auxiliary nodes to satisfy the topological constraints of low-dimensional planes and optimize planar configurations.

Its dynamic characteristics are reflected as follows: dispersed flow components in the high-dimensional depth space are gradually constrained, converged and redirected, with depth flux decaying in an orderly fashion. Controlled collapse of spatial dimensional degrees of freedom is achieved while fully preserving global topological connectivity and hierarchical branch structures.

1.2 Information-Theoretic Mechanism of Dimensionality Reduction

From an information-theoretic perspective, the continuous dimensionality reduction transformation proposed in this paper realizes ordered regularization and reconstruction of system information, rather than information annihilation and feature loss inherent to traditional dimensionality reduction approaches.

Via the global flow redirection mechanism, topological structural information originally scattered in the three-dimensional depth space gradually settles and converges into the two-dimensional planar space. While this mechanism cuts down spatial redundancy, structural complexity and computational degrees of freedom, it completely retains backbone network topology, hierarchical branch relations and core structural features, unifying geometric dimensional reduction and topological information preservation.

1.3 A New Axiom System for Structured Dimensionality Reduction

Based on topological evolution laws and information convergence mechanisms, this paper revises the cognition of dimensionality reduction in fractal geometry and engineering topology, and establishes an innovative paradigm distinct from destructive traditional dimensionality reduction:

\text{Dimensionality Reduction} \neq \text{Information Loss, Structural Damage, Feature Annihilation}

\text{Dimensionality Reduction} \triangleq \text{Controllable Flow Redistribution} + \text{Ordered Convergence of Spatially Dispersed Information}

The assertion that "dimensionality reduction inevitably causes information loss" arises from a one-sided understanding shaped under the traditional single-origin geometric framework. It only applies to truncated static projection approaches, instead of being a universal law governing cross-dimensional topological deformation.

This paradigm thoroughly departs from lossy structural-clipping methods including projective dimensionality reduction, discrete slicing and mesh simplification, and constructs a novel theoretical framework for continuous structured dimensionality reduction that preserves topology, connectivity and core features.

II. Fundamental Complementary Relationship with Shannon’s Information Theory

Shannon’s information theory constructs a quantitative framework for information uncertainty, defines theoretical limits for coding, transmission and compression, and addresses core issues of information quantification and capacity bounds. Nevertheless, its theoretical system is completely decoupled from spatial topology and transport geometric constraints, and cannot characterize morphological evolution and flow migration patterns of information within high-dimensional and low-dimensional spatial structures.

The theory put forward in this paper achieves rigorous fundamental complementarity with Shannon’s information theory: Shannon’s framework tackles the algebraic boundary problem concerning the "quantity" of information, while this paper addresses the geometric structural problem of information "spatial transport and path allocation". Together, they form a dual-layer fundamental support system of "algebraic quantification – geometric topology" for modern information science.

2.1 Dimensional Complementarity: Numerical Abstract System versus Spatial Geometric Carrier

- Shannon’s Information Theory: Free from spatial dimensional constraints, it is a pure probabilistic numerical system that delivers algebraic quantitative description of information;
- MOC Fractal Transformation System: Constrained by 2D/3D fractal topology and multi-origin spatial metrics, it realizes spatial geometric description of information transport.

Combined, the two systems fully cover the two core dimensions of information: numerical attributes and spatial transmission structures.

2.2 Complementary Compression Logic: Theoretical Lower Bound versus Lossless Topological Convergence

- Shannon’s Information Theory: Derives the theoretical lower bound of source compression. It only judges compression feasibility and ultimate compression scale, without providing geometric compression schemes that retain complete transmission paths;
- Theoretical Framework of This Paper: Provides engineering-feasible controllable geometric compression mechanisms. While approaching the lower bound of rate-distortion complexity, it leverages multi-origin redundant structures and 2-connected topological properties to ensure local faults cannot interrupt global transmission, resolving the problem of topological reliability during compression.

2.3 Complementary Engineering Chains: Signal Coding Standards versus Topological Design Criteria

- Shannon’s Theoretical System: Underlies signal-layer engineering standards such as communication coding, modulation schemes and transmission protocols;
- MOC Dimensional Gradient Transformation Framework: Supports topological design criteria for complex networks including three-dimensional Internet of Things, 3D chip interconnection, vascular topological simulation and spatial digital twins.

The two systems cover signal-layer coding rules and spatial-layer flow scheduling rules respectively, forming a complete theoretical support chain for distributed information systems.

III. The Novel Axiom System Established in This Paper

Axiom I — Topology Preservation Invariance

The dimensionality reduction transformation fully retains all connected components, loop structures and hierarchical relations throughout evolution. No structural information is lost in the process — only depth coordinates are discarded, while complete topological connotations migrate intact to the low-dimensional plane.

Axiom II — Topological Flux Conservation

The total global topological flow of the system acts as an invariant over evolution. The total volume of information remains strictly conserved during cross-dimensional deformation; only the spatial distribution pattern changes, with no fluctuation in total information quantity.

Axiom III — Limit Lossless Approximation

The structural complexity of the system can approach the theoretical lower bound permitted by Shannon’s rate-distortion theory. Unlike traditional lossy compression, which inevitably degrades structural quality when approaching compression limits, this paper adopts a topological self-reconstruction mechanism to maintain complete and intact structural information throughout dimensional convergence.