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arxiv: 2607.00852 · v1 · pith:6RJGLQIZnew · submitted 2026-07-01 · 💻 cs.CL · cs.AI

Recovering Input Text from Hidden States: Study of Gradient-Based Inversion of Decoder-Only Language Models

Miko{\l}aj S{\l}owikowski , Maciej Witold Majewski This is my paper

Pith reviewed 2026-07-02 13:23 UTC · model grok-4.3

classification 💻 cs.CL cs.AI
keywords hidden state inversiondecoder-only language modelstext recoverygradient-based optimizationGPT-2privacy leakageembedding space optimizationdiscrete loss
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The pith

Last-layer hidden states of GPT-2 recover the original input text through continuous embedding optimization.

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

The paper investigates recovering the exact input token sequence to a decoder-only language model solely from its final hidden states. It frames the task as an optimization that stays in continuous embedding space, driving a soft proxy toward the target state and committing to discrete tokens only at the end. This choice surfaces observable signals such as per-position loss curves, token rank trajectories, and a discrete loss value that can be summed to gauge overall recovery success. Experiments on short C4 prompts show that content words reconstruct almost perfectly while space-prefixed high-frequency function words account for nearly all failures. Widening the candidate window at commit time lifts exact-match accuracy from 66.9 percent to 97.5 percent, indicating that most errors are near-misses rather than true ambiguities.

Core claim

By keeping optimization entirely in continuous embedding space without any hard-token projection during search and committing tokens only once at the end, the last-layer hidden states of GPT-2 permit near-exact recovery of the input sequence; content-bearing tokens succeed almost always while space-prefixed high-frequency words in dense embedding regions cause the failures, and the resulting discrete loss serves as a reliable indicator of correctness.

What carries the argument

Continuous embedding-space optimisation of a soft proxy toward the leaked last-layer hidden-state target, with token commitment performed only after the inner search loop completes.

If this is right

  • Recovery correctness can be evaluated from the discrete loss alone, without access to the ground-truth tokens.
  • High-frequency function words prefixed by space and lying in dense embedding regions dominate reconstruction errors.
  • Content tokens are recovered with near-perfect fidelity under the same procedure.
  • Widening the candidate window at the final commit step converts most near-miss errors into exact matches.

Where Pith is reading between the lines

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

  • The same continuous-search signals could be used to compare sensitivity across different layers or model scales.
  • If hidden states are ever exposed, the demonstrated recovery rates imply concrete privacy leakage for typical user prompts.
  • The discrete-loss detector might be turned into a lightweight filter that flags or sanitizes high-risk hidden states before release.

Load-bearing premise

Keeping the optimization entirely in continuous space without hard-token projection during search produces reliable internal signals and a discrete loss that correctly indicates recovery quality.

What would settle it

Measure whether the cumulative discrete loss computed at commit time on held-out prompts accurately ranks the recoveries by their exact-match rate to the original text.

Figures

Figures reproduced from arXiv: 2607.00852 by Maciej Witold Majewski, Miko{\l}aj S{\l}owikowski.

Figure 1
Figure 1. Figure 1: Per-token continuous loss curves under Adam versus Adam with cosine [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Adam stabilisation: effect of cosine annealing and gradient clipping on [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Exact-match rate vs. step budget K (C = 2000, C4 prompts). Accuracy rises steeply up to K ≈ 1000, after which returns diminish. configuration terminates at step 600, so the Fast cut-off sacrifices no accuracy on them. For mis-committed tokens the median never approaches rank 1 regardless of step count: the optimiser fails to reach the correct neighbourhood for this minority, and the only remedy is a wider … view at source ↗
Figure 4
Figure 4. Figure 4: Rank of the ground-truth token at the first wrongly committed position [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Rank trajectory of the ground-truth token for the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cumulative discrete loss per token position (2000-prompt batch, loga [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Median continuous loss vs. optimisation step, grouped by token position [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: the 15 most frequently mis-committed ground-truth token strings [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

This work studies the hidden-state inversion problem: recovering the original input token sequence of a decoder-only language model from its last-layer hidden states. Rather than treating inversion as a one-shot reconstruction, we study it as a continuous embedding-space optimisation in which a soft proxy is driven towards the leaked target without any hard-token projection during the search, and a token is committed only once, at the end of the inner loop. This design choice has two consequences which are the main focus of this paper. First, keeping the optimisation entirely in continuous space exposes a rich set of internal signals: rank trajectories of the ground-truth token, per-position loss curves, and a discrete loss measured at commit time. Second, the discrete loss allows assessing the correctness of recovery via cumulative discrete loss. We further analyse which tokens break the reconstructions and find a sharp categorical asymmetry: space-prefixed, high-frequency function words in dense regions of the embedding matrix dominate the failures, while content-bearing tokens are recovered almost perfectly. On 10-token C4 prompts the exact-match rate rises from 66.9% to 97.5% (mean similarity 0.994) as the candidate window is widened, confirming that most errors are recoverable near-misses rather than genuine ambiguities. A comparison with the released SIPIT reference situates these findings: per-step hard projection is faster, but the continuous formulation is what makes the optimisation observable and its failures detectable. The results show that last-layer hidden states of GPT-2 are as sensitive as the original text.

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

2 major / 2 minor

Summary. The manuscript studies the hidden-state inversion problem for decoder-only language models (GPT-2) by casting recovery as continuous embedding-space optimization that avoids any hard-token projection until a single commit step at the end of the inner loop. This design is claimed to expose observable internal signals (rank trajectories, per-position loss curves, discrete loss at commit time) and to enable assessment of recovery quality via cumulative discrete loss. On 10-token C4 prompts the paper reports exact-match rates rising from 66.9% to 97.5% (mean similarity 0.994) when the candidate window is widened, attributes most errors to space-prefixed high-frequency function words, and contrasts the approach with SIPIT to argue that last-layer hidden states are as sensitive as the original text.

Significance. If the quantitative recovery rates and the claimed observability of the continuous formulation hold under full experimental controls, the work would demonstrate substantial information leakage from last-layer activations and supply a methodological tool for diagnosing inversion dynamics that discrete-projection baselines obscure. The categorical asymmetry finding (function-word failures versus near-perfect content-token recovery) would also contribute to understanding embedding-space geometry. No machine-checked proofs or parameter-free derivations are present; the contribution is empirical.

major comments (2)
  1. [Abstract / Experimental Setup] Abstract and experimental description: the central quantitative claims (66.9% and 97.5% exact-match rates, 0.994 similarity) are load-bearing for the sensitivity conclusion, yet no information is supplied on the number of prompts evaluated, train/test splits of C4, number of random seeds, optimization hyperparameters, or error bars. Without these details the reported rates cannot be assessed for statistical reliability.
  2. [Design Consequences] Design-consequences paragraph: the assertion that keeping optimization entirely in continuous space produces reliable internal signals and that the discrete loss at commit time correctly indicates recovery quality is load-bearing for the methodological novelty. The manuscript should supply an explicit correlation or ablation (e.g., scatter plot of discrete loss versus final exact-match accuracy across positions) rather than relying on qualitative description alone.
minor comments (2)
  1. A side-by-side table comparing SIPIT and the continuous method on both runtime and recovery metrics would make the positioning clearer.
  2. [Abstract] The phrase "as sensitive as the original text" in the final sentence of the abstract is informal; a more precise formulation (e.g., "recoverable with high fidelity under the reported optimization") would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will incorporate the requested details and analysis into a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / Experimental Setup] Abstract and experimental description: the central quantitative claims (66.9% and 97.5% exact-match rates, 0.994 similarity) are load-bearing for the sensitivity conclusion, yet no information is supplied on the number of prompts evaluated, train/test splits of C4, number of random seeds, optimization hyperparameters, or error bars. Without these details the reported rates cannot be assessed for statistical reliability.

    Authors: We agree that these experimental details are necessary for evaluating the reliability of the reported rates. The original manuscript omitted them due to length constraints. In the revision we will add: evaluation on 1000 randomly sampled 10-token prompts from the C4 validation split; results averaged over 3 random seeds with standard-deviation error bars; and the full optimization hyperparameters (learning rate, number of inner-loop steps, candidate-window sizes, and commit criterion). These additions will appear in a new Experimental Setup subsection. revision: yes

  2. Referee: [Design Consequences] Design-consequences paragraph: the assertion that keeping optimization entirely in continuous space produces reliable internal signals and that the discrete loss at commit time correctly indicates recovery quality is load-bearing for the methodological novelty. The manuscript should supply an explicit correlation or ablation (e.g., scatter plot of discrete loss versus final exact-match accuracy across positions) rather than relying on qualitative description alone.

    Authors: We accept that a quantitative demonstration would strengthen the claim. The current manuscript presents the signals and the discrete-loss indicator only qualitatively. In the revision we will add a scatter plot (and associated Pearson correlation) of per-position discrete loss at commit time versus exact-match accuracy, aggregated across all evaluated positions and prompts. This plot will be placed in the Design Consequences section to provide the requested explicit evidence. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents an empirical study of gradient-based inversion via continuous-space optimization on GPT-2 last-layer hidden states. The central claim (last-layer states are as sensitive as original text) rests on reported exact-match rates (66.9% to 97.5%), similarity scores, and categorical failure analysis, none of which reduce to fitted parameters, self-definitions, or self-citation chains. The continuous-space design is introduced as an explicit methodological choice whose consequences (observable signals, discrete loss) are then measured; no derivation equates outputs to inputs by construction. No equations appear in the abstract or described content, and the SIPIT comparison is external. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are introduced or quantified. The work relies on standard gradient optimization assumptions in embedding space.

axioms (1)
  • domain assumption Gradient descent in continuous embedding space can be driven toward a leaked target hidden state without intermediate discrete projections
    Core design choice stated in the abstract as enabling observable signals and final discrete loss measurement.

pith-pipeline@v0.9.1-grok · 5820 in / 1128 out tokens · 28086 ms · 2026-07-02T13:23:06.998365+00:00 · methodology

discussion (0)

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