Understanding and Insights on “Language Models Are Injective and Hence Invertible”

📊 Core Four-Element Analysis

1️⃣ Fundamental Problem

The field widely assumes that Transformer language models are "lossy" due to non-linear activations, normalization, and many-to-one attention mechanisms—meaning different inputs could collapse to identical hidden states, making exact input recovery impossible. This assumption limits our understanding of model transparency, interpretability, and safe deployment.

2️⃣ Key Insight

The authors approach this from real analysis, treating Transformers as real-analytic functions over parameter space. Leveraging the mathematical property that real-analytic functions have zero sets of measure zero, they prove that "collisions" (different inputs producing identical representations) can only occur on a measure-zero set of parameters—a mathematical exception rather than a practical possibility.

3️⃣ Key Method

Method Formula:

Injectivity Guarantee = (Real-analyticity × Zero-set Measure Zero) 
                       + (Absolute Continuity of Standard Init) 
                       + (GD Preserves Absolute Continuity)

Implementation Path:

Theoretical Proof (Three Steps):

1. Establish Real-analyticity: Prove every Transformer component (embeddings, LayerNorm, causal attention, MLPs, residuals) is real-analytic in parameters, hence the entire model is real-analytic
2. Injectivity at Initialization: For any two distinct prompts $s \neq s'$, construct collision function $h(\theta) = \|r(s;\theta) - r(s';\theta)\|^2_2$, prove $h$ is not identically zero, therefore its zero set has measure zero; standard initialization (Gaussian/uniform) has density, almost surely avoids zero set
3. Training Preserves Injectivity: Prove each gradient descent step $\phi(\theta) = \theta - \eta\nabla L(\theta)$ has Jacobian determinant non-zero almost everywhere; by Inverse Function Theorem, $\phi$ preserves absolute continuity of parameter distribution, so training never moves parameters into collision set

Empirical Validation (Three Levels):

1. Large-scale Collision Search: 5 billion pairwise comparisons across 6 SOTA models, zero collisions found
2. Exhaustive Stress Test: For 10 closest prompt prefixes, tried all vocabulary token continuations (343 billion comparisons), still no collisions
3. Invertibility Algorithm SIPIT: Designed linear-time algorithm for exact input reconstruction from hidden states, 100% accuracy validates theory

4️⃣ Core Findings

1. Theoretical Guarantee: Under standard initialization and finite-step gradient training, decoder-only Transformers are almost surely injective—different prompts produce different last-token representations (probability 1)

2. Empirical Confirmation: Across billions of collision tests on GPT-2, Gemma-3, Llama-3.1, Mistral, minimum L2 distances far exceed collision threshold (10^-6), and distances increase with depth

3. Operationalizability: SIPIT algorithm achieves linear-time ($O(T|V|)$ worst case) exact prompt reconstruction, averaging 28 seconds vs. 3890 seconds for brute force, with 100% accuracy

4. Depth Dependency: Hidden state separability increases with layer depth, indicating deeper representations retain richer input information

5. Failure Boundaries: Collisions can only be manufactured through deliberate non-analytic choices (quantization, weight tying, non-smooth activations)

🔬 Method Formula (Extended)

Mathematical Framework

Injectivity ⟺ ∀s ≠ s' : r(s;θ) ≠ r(s';θ)

Proof Path:
1. Real-analyticity: Transformer ∈ C^ω(parameter space)
2. Collision function: h(θ) = ||r(s;θ) - r(s';θ)||²
3. Zero set: Lebesgue measure of {θ : h(θ) = 0} = 0
4. Initialization: P(θ₀ ∈ zero set) = 0  (density exists)
5. Training preserves: det(Dφ(θ)) ≠ 0 a.e. ⟹ absolute continuity preserved

Algorithm Framework (SIPIT)

Input: Observed hidden states Ĥ^(ℓ) ∈ R^(T×d)
Output: Reconstructed sequence ŝ = ⟨ŝ₁,...,ŝ_T⟩

for t = 1 to T:
    for each candidate token v_j ∈ V:
        compute h_t(prefix ⊕ v_j)
        if ||ĥ_t - h_t(prefix ⊕ v_j)||₂ < ε:
            ŝ_t ← v_j
            break

Time Complexity: O(T|V|) worst case, faster in practice (gradient-guided policy)

💡 Final Dual Summary

One-Sentence Summary (Core Value)

This work rigorously proves via real analysis that standard decoder-only Transformers are almost surely injective under all practical parameter settings and training procedures—different inputs necessarily produce different last-token representations—and introduces SIPIT, an algorithm that reconstructs exact inputs from hidden states in provable linear time, elevating "Transformer representations are lossless" from longstanding assumption to mathematical theorem and operational engineering tool, providing a new theoretical foundation for model transparency, interpretability, and privacy security.

One-Sentence Summary (Plain English)

Like fingerprints, this paper mathematically proves that whenever you feed AI different text, its "brain" (hidden layers) produces guaranteed-unique "fingerprints," and we can 100% recover the original text from these fingerprints—meaning AI's "memory" is far more complete than we thought: even after deleting input text, its internal representations still perfectly preserve all information.

Reference:

Paper: https://www.arxiv.org/abs/2510.15511