TFZ: Topology-Preserving Tensor Compression

• TFZ is a framework that preserves topology in 2D symmetric and asymmetric tensor fields on flat triangular meshes by correcting cell-wise distortions.
• It maintains critical invariants such as degenerate points, wedges, trisectors, and eigengraph connectivity for reliable downstream analysis and visualization.
• TFZ integrates with error-bounded compressors like SZ3 and SPERR, applying quantization and local correction techniques to enforce original tensor topology.

Searching arXiv for the provided TFZ papers to ground the article and handle acronym disambiguation. TFZ most commonly denotes a topology-preserving lossy compression framework for 2D symmetric and asymmetric second-order tensor fields defined on flat triangular meshes. In that sense, TFZ is designed to preserve topological structure under compression, specifically degenerate points in symmetric tensor fields and eigenvector and eigenvalue graph structure in asymmetric tensor fields, by enforcing cell-wise topological invariants during a post-decompression correction stage (Gorski et al., 12 Aug 2025). The acronym is also used in a distinct later context, TFZ-Tree, where “TFZ” stands for “Time–Frequency Z-test” in an ultra-lightweight waveform classification framework for resource-constrained devices (Wang et al., 15 May 2026). In current usage, the tensor-field compression framework is the primary referent when TFZ is discussed in computational topology, visualization, and scientific data compression.

1. Definition and scope

TFZ, in the sense introduced in "TFZ: Topology-Preserving Compression of 2D Symmetric and Asymmetric Second-Order Tensor Fields" (Gorski et al., 12 Aug 2025), is a compression framework that preserves the topology of 2D symmetric and asymmetric second-order tensor fields on flat triangular meshes. A tensor field assigns a tensor to each point in space; the paper explicitly situates such fields in applications including stress and strain tensors, the Riemann curvature tensor, graphics, and neuroscience. The central problem addressed is that lossy compression may distort tensor-field topology, thereby hindering downstream analysis and visualization tasks.

The framework is defined as operating on top of an error-bounded compressor. The compressor first produces an intermediate decompressed field, and TFZ then scans each triangular cell and corrects local topological violations. The stated design principle is local-to-global: TFZ preserves the local topology of each cell and thereby ensures certain global topological guarantees (Gorski et al., 12 Aug 2025).

A separate use of the acronym appears in "TFZ-Tree: An Ultra-Lightweight Waveform Classification Framework for Resource-Constrained Devices" (Wang et al., 15 May 2026). There, TFZ stands for “Time–Frequency Z-test,” combining time-frequency feature engineering with a Z-test-based decision tree for waveform classification. This is terminologically unrelated to tensor-field compression, and the coexistence of the two usages makes disambiguation necessary in bibliographic and technical contexts.

2. Mathematical setting for 2D second-order tensor fields

In the tensor-compression framework, the domain is a flat triangular mesh $X \subset \mathbb{R}^2$, and each vertex $v \in X$ stores a $2 \times 2$ tensor (Gorski et al., 12 Aug 2025). In the symmetric case,

$$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$

In the asymmetric case,

$$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$

On each triangle $\sigma=\triangle v_1v_2v_3$, each tensor component is linearly interpolated: $$T(a_1v_1+a_2v_2+a_3v_3) =a_1T(v_1)+a_2T(v_2)+a_3T(v_3),\quad\sum a_i=1,\;a_i\ge0.$$ This piecewise-linear formulation determines the cell-wise topology that TFZ attempts to preserve.

For symmetric fields, the relevant topological singularities are degenerate points, defined as points where the tensor has equal eigenvalues, $\lambda_1=\lambda_2$. Equivalently, the deviator

$D(T)\;=\;T-\tfrac12\mathrm{tr}(T)\,I =\begin{pmatrix}\Delta&F\F&-\Delta\end{pmatrix}$

vanishes, with $\det D(T)=0$ (Gorski et al., 12 Aug 2025). In each triangle, there may be at most one isolated degenerate point. Its local type is determined by the sign-tests

$v \in X$0

yielding a wedge if all $v \in X$1 and a trisector if all $v \in X$2.

For asymmetric fields, the formulation follows a Zhang–Pang decomposition into isotropic $v \in X$3, rotation $v \in X$4, anisotropic stretching $v \in X$5, and angle $v \in X$6. The dual-eigenvector field is the eigenvector field of the symmetric part

$v \in X$7

and its degenerate points coincide with those of $v \in X$8. Two manifold constructions organize the global topology: the eigenvalue manifold,

$v \in X$9

which is a hemisphere, and the eigenvector manifold,

$2 \times 2$0

which is a sphere (Gorski et al., 12 Aug 2025).

3. Topological invariants preserved in symmetric tensor fields

For symmetric tensor fields, TFZ preserves, for every cell, whether it contains no degenerate point, a single wedge, a single trisector, a degenerate edge, or an entire cell of degeneracy, and, when degeneracy lies on an edge, the precise location on that edge (Gorski et al., 12 Aug 2025). This is a stronger statement than merely preserving a global count of singularities; it enforces a cell-wise invariant from which the global structure is reconstructed.

The operational criterion is the local sign pattern $2 \times 2$1. The global set of degenerate points is exactly those cells whose local sign-patterns indicate a wedge or trisector. By forcing each cell’s local pattern to match the original, TFZ recovers exactly the same degenerate points and their classification. The paper further states that the connectivity of tensorlines, which connect wedges and trisectors, is determined by adjacency of these cells; since no new degenerate points appear and none disappear, the global skeleton is unchanged (Gorski et al., 12 Aug 2025).

This local characterization is significant because it converts a global topology-preservation problem into a sequence of per-cell predicate checks and corrections. A plausible implication is that the framework is particularly well matched to piecewise-linear tensor data on triangulated domains, where cell-local interpolation formulas already define the admissible topological events.

4. Topological invariants preserved in asymmetric tensor fields

For asymmetric tensor fields, TFZ preserves two graph structures derived from dominance partitions in the $2 \times 2$2-coordinates: the eigenvalue graph and the eigenvector graph (Gorski et al., 12 Aug 2025). The mesh is partitioned according to which coordinate dominates in absolute value, together with its sign, and the adjacencies between regions form the two graphs. TFZ preserves, cell by cell, both the partition inside each triangle and hence the global dual-eigenvector degenerate points plus the two graph structures.

Within each triangle, internal partition boundaries are traced by conics. The relation $2 \times 2$3 splits the eigenvector partition, while $2 \times 2$4, $2 \times 2$5, and $2 \times 2$6 split the eigenvalue partition. The framework records which edges these conics intersect, in what orientation, and any junction points; these features constitute the cell-wise invariant compared against the ground truth (Gorski et al., 12 Aug 2025).

The global guarantee is again expressed in local-to-global form. The eigenvalue and eigenvector graphs are the dual graphs of the cell-wise partition. TFZ enforces that each cell has the same partition polygon topology as the original, including the same number of regions and the same adjacencies along edges and junctions. Gluing cells together then recovers exactly the same global adjacency graphs (Gorski et al., 12 Aug 2025).

This asymmetric case extends the framework beyond symmetric eigendirection singularities to a richer set of invariant structures. The paper’s formulation suggests that, for asymmetric tensor fields, preserving graph connectivity is as important as preserving isolated singular points, because base compressors may otherwise break entire graph connectivity.

5. Cell correction algorithm and integration with compressors

TFZ is explicitly layered on top of any error-bounded compressor, and the paper demonstrates integration with SZ3 and SPERR (Gorski et al., 12 Aug 2025). The workflow has three stages. First, each tensor-component field is compressed separately with SZ3 or SPERR under a user-specified absolute error bound $2 \times 2$7, then decompressed to an intermediate field $2 \times 2$8. Second, TFZ runs a cell-correction pass. Third, all integer side-data are Huffman-encoded, combined with the compressed component streams and any lossless tensors into a tar archive, and the final bundle is compressed with zstd.

In the asymmetric case, the correction stage begins with vertex correction. For each vertex, TFZ reads $2 \times 2$9, compares signs and magnitudes to the ground-truth predicate, and, if a mismatch is found, applies one or more of the following operations: sign-fix by adding or subtracting $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$0, magnitude-swap when coefficient ordering is wrong, quantization of $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$1 when the dual-eigenvector degenerate type is incorrect, and flag storage so that the decoder can invert the transformation (Gorski et al., 12 Aug 2025). Internal topology correction then compares the conic-induced partition to the ground-truth invariant; if they differ, TFZ tightens the local error bound using logarithmic quantization or stores the entire cell losslessly if too many attempts fail.

For the symmetric case, the paper provides a pseudocode sketch in which, for each cell, TFZ computes deviators of the decompressed vertex tensors, evaluates the sign tests $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$2, compares them with ground truth, and, if a mismatch exists, quantizes the angular variable at the vertex whose $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$3 differs most from $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$4. The sketch allows repeated correction attempts at up to three vertices before falling back to lossless storage (Gorski et al., 12 Aug 2025).

Two quantization modes are specified. Linear scaling shifts the intermediate guess by $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$5 to guarantee $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$6. Logarithmic scaling assigns a per-point precision $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$7, allowing a variable local bound $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$8 (Gorski et al., 12 Aug 2025). The use of side-data and selective lossless fallback indicates that TFZ trades a bounded bitrate increase for exact recovery of the designated topological invariants.

6. Experimental results, performance, and acronym disambiguation

The reported experiments cover four symmetric datasets—Stress A/B and Brain A/B—and four asymmetric flow-derived datasets—Ocean, Miranda turbulence, Vortex Street, and Heated Cylinder (Gorski et al., 12 Aug 2025). With $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{12}(v)&T_{22}(v) \end{pmatrix},\quad T_{ij}=T_{ji}.$$9 for symmetric data and $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$0 for asymmetric data, augmented SZ3 and SPERR produce $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$1 misclassified cells per slice. At the same compressed size, base SZ3 and SPERR misclassify up to $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$2 of symmetric cells and $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$3 of asymmetric cells; the paper associates these errors with removal or insertion of degenerate points in the symmetric case and with broken graph connectivity in the asymmetric case.

Rate–distortion results are described as placing A-SZ3 and A-SPERR only slightly to the right of the corresponding base curves. The overhead is reported as $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$4 extra bits for symmetric fields and $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$5 extra bits for asymmetric fields when preserving both eigenvalue and eigenvector graphs. If only one manifold is preserved, the overhead drops significantly (Gorski et al., 12 Aug 2025).

Runtime overhead is also quantified. Base SZ3 on a single slice takes approximately $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$6 s, while augmented SZ3 adds a factor of $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$7 per slice, with most time spent in cell correction. Throughput is approximately $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$8 MB/s on a current 12-core CPU in Julia, and decompression overhead is described as mild, at approximately $$T(v)=\begin{pmatrix} T_{11}(v)&T_{12}(v)\ T_{21}(v)&T_{22}(v) \end{pmatrix}, \quad T_{21}\neq T_{12}\;\text{in general}.$$9 slower (Gorski et al., 12 Aug 2025). Visual examples on Stress B, Brain B, Miranda, and Ocean show that base compressors may omit or introduce wedges and trisectors or create spurious region islands, whereas the augmented variants recover the original partition and topology.

The acronym TFZ should, however, be read contextually. In TFZ-Tree, “TFZ” denotes “Time–Frequency Z-test,” not topology-preserving tensor compression (Wang et al., 15 May 2026). That framework operates on blocks of $\sigma=\triangle v_1v_2v_3$0 complex samples, extracts an 80-dimensional time-frequency feature vector, and classifies ten 6G candidate waveforms with a Z-test-optimized decision tree. Its reported average accuracy is $\sigma=\triangle v_1v_2v_3$1 under AWGN and $\sigma=\triangle v_1v_2v_3$2 under 3GPP TDL-C multipath, with single inference latency under $\sigma=\triangle v_1v_2v_3$3 ms on an x86 platform (Wang et al., 15 May 2026). The coexistence of these two meanings is a matter of nomenclature rather than methodological overlap.

A plausible implication is that “TFZ” is now a genuinely polysemous technical acronym: in scientific visualization and topology-aware compression it refers to a tensor-field framework, while in signal classification it names a time-frequency and Z-test pipeline. In current arXiv usage, the surrounding domain vocabulary—tensor fields, degenerate points, eigenvalue graphs, or, alternatively, waveform recognition and Z-test trees—determines the intended meaning.