Honest Physical-Support Inference after Latent Dictionary Learning: Collision Singularities and Minimax Resolution

Technical Summary: Honest Physical-Support Inference after Latent Dictionary Learning

1. Problem Formulation

The paper addresses the statistical challenge of inferring the active physical components (support) of a sparse signal when the dictionary of atoms has been learned from a separate, unlabeled training set of latent sparse mixtures.

• The Core Conflict: Standard sparse inference methods condition on a fixed dictionary. However, when a dictionary is learned from latent data, it possesses an inherent "orientation" ambiguity near coherent atom collisions. Specifically, near a collision of coherent atoms (a "coherent block"), the training data may identify the active group of atoms but fail to resolve the specific physical rays (children) within that group due to rotational symmetries in the latent space.
• The "Plug-in" Failure: Freezing a single learned dictionary and treating it as known converts this unresolved calibration uncertainty into a precise but potentially misleading statement about specific atom labels. The paper argues that a "known-dictionary" analysis can distinguish coordinate supports while the training experiment cannot identify the physical rays to which those coordinates refer.
• The Goal: The objective is to construct an "honest" inference procedure that outputs the finest sparse statement justified by the joint train–test experiment. The target is not a coordinate support in an arbitrary frame, but a set of active physical rays (modulo simultaneous dictionary-support relabeling) that remains compatible with both the training data and the test signal.

2. Methodology

The proposed method avoids selecting a single "best" dictionary. Instead, it employs a cross-dictionary confidence correspondence that profiles the test data against a region of all dictionaries compatible with the training data.

• Training Phase (Observable Invariants):

  • The method utilizes second-order ($M_2$) and fourth-order ($K_4$) moments of the training data.
  • Crucially, it identifies that while second-order moments determine the block shape, fourth-order moments are required to recover residual physical orientation. The paper proves that orientation information enters the latent training density at cubic order, yielding an information budget of order $N s^6$ (where $N$ is the training sample size and $s$ is the collision scale).
  • A robust confidence region for these moments is inverted to define a set of all "training-compatible" dictionary orbits (modulo child permutations).

• Test Phase (Joint Profiling):

  • For a test signal observed $T$ times, the method profiles the test mean against every dictionary in the training-compatible region.
  • It constructs a joint confidence region for the dictionary, the support set, and the coefficients.
  • The final output is the projection of this joint region onto the physical-support space (the set of active physical rays).

• Output Structure:
  The procedure does not force a single leaf-level answer. Instead, it produces a set-valued output that adapts to the available information:

  1. Cross-sheet inconclusive: If the data cannot distinguish between the coherent block and a separated alternative.
  2. Group-resolved but child-ambiguous: If the coherent parent is identified, but the specific active children within the block cannot be resolved.
  3. Fine-support resolved: If both the parent and the specific children are resolved.

3. Key Contributions and Theoretical Results

C1. Collision-Singular Training Geometry
The paper establishes a statistical geometry for latent sparse training near atom collisions. It proves that the "all-pair" Hellinger/Kullback–Leibler chord is regular in observable invariants, but the physical inverse loses powers of the collision scale $s$. Specifically, the orientation information is of order $s^6$, meaning that resolving the physical rays requires $N s^6$ information, not the standard $N$.

C2. Honest Cross-Dictionary Confidence Correspondence
The authors construct a finite-sample confidence correspondence with a two-level honesty guarantee (Equation 3). This guarantee ensures that, with high probability over the training data, the conditional test-time coverage of the physical support is high. The method requires no observed training codes and no support-labeled calibration samples.

C3. Three-Gate Diameter Bound and Minimax Optimality
The resolution of the output is governed by three separate information budgets:

1. Parent Detectability ($I_G$): Distinguishing the coherent block from a separated anchor.
2. Test-Time Support Separation ($I_S$): Distinguishing specific child supports within the block.
3. Learned-Dictionary Orientation ($I_D = N s^6$): Resolving the physical rays from the training data.

The paper proves a minimax lower bound showing that these three gates are unavoidable. In the resolved regime (where parent and support gates are open), the projective Hausdorff diameter of the confidence set contracts at the rate $s \wedge (N s^2)^{-1}$, which is optimal.

C4. Task-Dependent Symmetry Breaking
The paper characterizes when additional test data can compensate for poor dictionary training. It shows that if the test coefficients are symmetric (e.g., equal amplitudes), the test mean remains invariant under certain dictionary rotations, meaning no amount of test replication can resolve the physical rays. However, if there is coefficient asymmetry (amplitude contrast), the test mean changes transversely, allowing test data to supplement the training information. The paper provides an explicit crossover condition for this effect.

4. Significance and Claims

The paper claims to provide a sharp physical-support diameter through a collision-singular learned representation. Its primary significance lies in:

• Honesty over Precision: It rejects the practice of forcing a precise but unstable atom label when the dictionary is uncertain. Instead, it provides a resolution-adaptive statement that explicitly acknowledges when the data only justify a group-level conclusion.
• Information Budgets: It identifies the specific statistical price of latent dictionary learning, showing that orientation information is of order $s^6$ rather than the standard $N$. This explains why "more test data" does not automatically fix dictionary uncertainty unless the test signal has specific asymmetries.
• Conjunction of Tools: The contribution is not a new algorithm for dictionary learning or a new group inference method, but the conjunction of group inference, honest sparse confidence sets, and orbit recovery in a latent train–test setting.

The paper explicitly states it does not claim to be the first in any of its component areas (e.g., group inference, set-valued prediction, or dictionary identifiability). Rather, it establishes the exact conjunction of these tools to solve the specific problem of physical-support inference after latent dictionary learning, providing a minimax-optimal resolution limit for this joint experiment. The results are currently limited to fixed-dimensional settings and a single supplied coherent block, serving as a theoretical benchmark for understanding the trade-offs between training calibration and test replication.