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arxiv: 2501.05958 · v3 · submitted 2025-01-10 · 🧮 math.NA · cs.NA· physics.chem-ph· physics.comp-ph

Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions

Yuyang Wang , Yukuan Hu , Xin Liu This is my paper

Pith reviewed 2026-05-23 05:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.chem-phphysics.comp-ph
keywords antisymmetric tensor product functionsrepresentation complexitycanonical polyadic ranklower boundshigh-dimensional problemsquantum many-body systemsnumerical analysis
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The pith

The minimum number of terms for a class of tensor product functions to be exactly antisymmetric grows exponentially with dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that a class of tensor product functions (TPFs) cannot achieve exact antisymmetry unless the number of terms grows exponentially with the problem dimension. This bound applies equally to discretized TPFs and to versions parameterized by neural networks. A reader would care because TPF approximations are promoted for high-dimensional PDEs and eigenvalue problems on the basis of linear scaling, yet the antisymmetry constraint central to quantum many-body systems forces an exponential cost. The argument proceeds by identifying the TPFs with antisymmetric tensors and lower-bounding their Canonical Polyadic rank.

Core claim

The minimum number of involved terms for a class of TPFs to be exactly antisymmetric increases exponentially fast with the problem dimension. This class encompasses both traditionally discretized TPFs and recent neural-network-parameterized ones. The proof exploits the link between the antisymmetric TPFs and the corresponding antisymmetric tensors, focusing on the Canonical Polyadic rank of the latter, and concludes that low-rank TPFs are fundamentally unsuitable for high-dimensional problems where antisymmetry is essential.

What carries the argument

The identification of antisymmetric TPFs with antisymmetric tensors whose Canonical Polyadic rank supplies the exponential lower bound on the number of terms.

If this is right

  • Low-rank TPFs cannot exactly represent antisymmetric functions without incurring exponential cost in high dimensions.
  • The same exponential lower bound applies to neural-network-parameterized TPFs as to classical discretized ones.
  • TPF approximations lose their advertised linear scaling once antisymmetry is required.
  • High-dimensional quantum many-body problems lie outside the regime where low-rank TPFs can be exact.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Representations that enforce antisymmetry by construction, rather than through the choice of terms, may evade the exponential barrier identified here.
  • Analogous rank lower bounds may exist for other symmetry constraints, such as symmetry under particle exchange or under spatial rotations.
  • Practical approximations would need to quantify the error incurred when antisymmetry is only approximately satisfied by low-rank TPFs.

Load-bearing premise

The minimum number of terms required by the TPF class equals or exceeds the Canonical Polyadic rank of the associated antisymmetric tensor, and this identification holds for both discretized and neural-network cases.

What would settle it

An explicit construction, for arbitrarily large dimension, of an antisymmetric tensor whose Canonical Polyadic rank grows only polynomially, or a concrete antisymmetric TPF with polynomially many terms that satisfies the antisymmetry condition exactly.

Figures

Figures reproduced from arXiv: 2501.05958 by Xin Liu, Yukuan Hu, Yuyang Wang.

Figure 1
Figure 1. Figure 1: Numerical comparison of the TNN approximations with and without explicit antisym [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of TNN. TNN wave function ansatz follows a similar structure. Each electronic coordinate is represented by a d-dimensional vector rj = (rj1, . . . , rjd) ⊤ ∈ R d , with each one-dimensional coordinate processed by an independent subnetwork, resulting in a total of dN independent subnetworks [34]. The TNN wave function ansatz for this system is fθ(r) := Xp i=1 YN j=1 Yd k=1 ψijk(rjk; θjk) ! , ∀… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of TNN approximations with and without antisymmetrization for the one [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

Tensor product function (TPF) approximations have been widely adopted in solving high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving desirable accuracy with computational overhead that scales linearly with problem dimensions. However, recent studies have underscored the extraordinarily high computational cost of TPFs on quantum many-body problems, even for systems with as few as three particles. A key distinction in these problems is the antisymmetry requirement on the unknown functions. In the present work, we rigorously establish that the minimum number of involved terms for a class of TPFs to be exactly antisymmetric increases exponentially fast with the problem dimension. This class encompasses both traditionally discretized TPFs and the recent ones parameterized by neural networks. Our proof exploits the link between the antisymmetric TPFs in this class and the corresponding antisymmetric tensors and focuses on the Canonical Polyadic rank of the latter. As a result, our findings reveal that low-rank TPFs are fundamentally unsuitable for high-dimensional problems where antisymmetry is essential.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove that the minimal number of terms in a class of tensor product functions (TPFs) needed to achieve exact antisymmetry grows exponentially with dimension. The class is asserted to include both basis-discretized TPFs and neural-network-parameterized TPFs. The argument proceeds by identifying antisymmetric TPFs with antisymmetric tensors and invoking a lower bound on their Canonical Polyadic (CP) rank.

Significance. If the claimed equivalence between term count and CP rank holds for both discretization and NN cases, the result supplies a concrete exponential lower bound that explains why low-rank TPF representations become impractical for antisymmetric high-dimensional problems such as quantum many-body systems. The manuscript does not supply machine-checked proofs or reproducible code, but the derivation is presented as parameter-free once the tensor identification is granted.

major comments (1)
  1. [Proof of the main result (Section 3 / Theorem 1)] The central reduction equates the minimal TPF term count to the CP rank of the associated antisymmetric tensor. This identification is immediate when univariate factors are expanded in a fixed finite basis (discretized case). For NN-parameterized univariate maps, which are arbitrary nonlinear functions, no explicit reduction is supplied showing that any such NN-TPF can be represented or bounded by a finite-basis expansion whose rank is controlled by the term count. Without this step the exponential lower bound does not automatically transfer to the NN setting, which is load-bearing for the claim that the result covers “both traditionally discretized TPFs and the recent ones parameterized by neural networks.”
minor comments (2)
  1. [Introduction] The precise definition of the “class of TPFs” under consideration should be stated explicitly in the introduction rather than deferred to the proof section.
  2. [Preliminaries] Notation for the antisymmetric tensor and its CP decomposition should be introduced with a short self-contained paragraph before the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. The main issue raised concerns the applicability of the lower bound to neural network-parameterized tensor product functions. We provide a response below and will update the manuscript to include the missing details.

read point-by-point responses

  1. Referee: [Proof of the main result (Section 3 / Theorem 1)] The central reduction equates the minimal TPF term count to the CP rank of the associated antisymmetric tensor. This identification is immediate when univariate factors are expanded in a fixed finite basis (discretized case). For NN-parameterized univariate maps, which are arbitrary nonlinear functions, no explicit reduction is supplied showing that any such NN-TPF can be represented or bounded by a finite-basis expansion whose rank is controlled by the term count. Without this step the exponential lower bound does not automatically transfer to the NN setting, which is load-bearing for the claim that the result covers “both traditionally discretized TPFs and the recent ones parameterized by neural networks.”

    Authors: We appreciate the referee pointing out the need for an explicit argument in the NN case. To clarify: consider any TPF representation with k terms that is exactly antisymmetric. Discretizing the domain using the same finite grid for each variable produces an antisymmetric tensor. The discretized TPF corresponds to a CP decomposition of this tensor with at most k terms, hence the CP rank is at most k. However, it is known (and used in our proof for the discretized case) that the CP rank of nonzero antisymmetric tensors grows exponentially with the dimension. This yields the same exponential lower bound on k for the NN case as well. We will insert this reduction explicitly into the revised Section 3 to strengthen the presentation. revision: yes

Circularity Check

0 steps flagged

No circularity detected; lower bound derived from external CP-rank definition

full rationale

The derivation establishes a lower bound on the number of TPF terms required for exact antisymmetry by linking the TPF class to the CP rank of the associated antisymmetric tensor. This link is presented as a mathematical equivalence based on the standard definition of CP rank (minimal number of rank-1 terms), which is an independent external concept rather than a fitted parameter or self-referential definition. The abstract and description indicate the proof applies the same tensor-rank argument to both discretized and NN-parameterized cases without any reduction of the claimed bound to the input assumptions by construction. No self-citations, ansatzes, or renamings are identified as load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mapping from antisymmetric TPFs to antisymmetric tensors and the properties of their CP rank; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Antisymmetric TPFs in the considered class correspond to antisymmetric tensors whose CP rank governs the minimal number of terms
    The proof exploits this link as stated in the abstract.

pith-pipeline@v0.9.0 · 5711 in / 1094 out tokens · 23361 ms · 2026-05-23T05:54:37.650336+00:00 · methodology

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