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A proximal method may reveal a smooth active manifold after a finite number of iterations. The useful question is not only whether the current support looks stable, but when the proof permits the algorithm to treat that structure as final and activate a faster local step.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the evidence
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE EVIDENCE
1. Identification time is a mathematical event
An active manifold represents the locally stable structure of a nonsmooth problem: a support pattern, active face or fixed-rank set. Identification time is the first iteration after which the iterates lie on that manifold and remain there under the theorem’s hypotheses.
A numerical detector can suggest that the event occurred. It cannot replace partial smoothness, nondegeneracy, convergence and the algorithm-specific argument that guarantees finite entry and containment.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Separate detection from certification
Detection uses observable quantities such as a repeated support, a proximal active set, normal residuals and tangent consistency. Certification adds the assumptions that make those diagnostics logically sufficient.
Keep three labels: candidate transition, certified identification and post-identification acceleration. This prevents a plotting convention or zero tolerance from being reported as a theorem.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. An iteration ledger
Consider a sparse composite problem whose solution has active support {1,3}. The ledger separates the visible pattern from the evidence needed to switch methods.
| Iterations | Observed support | Residual signal | Status |
|---|---|---|---|
| 1–7 | Changes repeatedly | Large normal residual | Pre-identification |
| 8–11 | {1,3,4} ↔ {1,3} | Boundary crossings | Candidate transition |
| 12–16 | {1,3} | Normal residual falls | Stable diagnostic |
| 17 onward | {1,3} | Containment test passes | Certified under stated theorem |
Iteration 12 is not automatically the identification time just because the support stops changing in this trace. The certificate records why leaving the manifold is excluded after iteration 17.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. A worked transition detector
Use three gates: the active label is unchanged for five accepted iterates; the normal residual is below 10⁻⁶; and a one-step prediction with the current local model returns to the same manifold. Passing all three flags a candidate switch.
| Gate | Observed value | Rule | Result |
|---|---|---|---|
| Support window | 5 unchanged | At least 5 | Pass |
| Normal residual | 6×10⁻⁷ | Below 10⁻⁶ | Pass |
| Predicted next manifold | Same {1,3} | Must agree | Pass |
| Theorem hypotheses | Checked separately | All required | Certificate condition |
If the Newton trial is rejected or leaves the manifold, return to the globally convergent proximal step and reset the detector.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Why transitions can be delayed
Near-degeneracy makes the normal signal small and can cause repeated boundary crossings. Approximate subproblem solves, stochastic gradients and changing step sizes may also delay exact entry or prevent it entirely.
The identification time can be large even when objective values look converged. A flat objective plot does not prove structural stability, and a late active-coordinate change can alter the correct reduced Hessian.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Post-manifold acceleration
After identification, restrict the smooth part of the objective to the active manifold and compute tangent gradients and a reduced Hessian or quasi-Newton model. Retractions keep the trial point on the manifold when straight ambient steps do not.
Superlinear acceleration needs more than identification: local regularity, a suitable second-order model, controlled inexactness and accepted steps are typical ingredients. The global proximal phase remains the safety mechanism outside that region.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: fixed-rank matrices
For a nuclear-norm problem, the active manifold may be a fixed-rank set. A proximal stage reveals candidate zero singular values; after certified identification, a Riemannian Newton-like step can work on the fixed-rank geometry.
The transfer check examines spectral gaps, repeated singular values and intersections with other constraints. A tiny singular value is not automatically zero, and nontransverse geometry can invalidate a naive reduced model.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Design a safe switching rule
Require sufficient decrease in the original objective, not only the reduced model. Use a trust region or line search, retain a fallback proximal step and count unsuccessful accelerations as evidence that identification or local regularity is incomplete.
Do not estimate a fast rate from the same short tail used to choose the switch and then call it a certificate. Predeclare the diagnostic window, residual and acceptance rule, and report the first qualifying iteration.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Connect the active-geometry branch
The stratified critical-set guide explains why rates can change across manifolds. The metric-subregularity guide supplies local error-bound logic.
Use variable KL geometry when the local modulus changes along the critical set rather than becoming uniform immediately.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Limits and evidence checks
Finite identification is not universal. Noise can keep iterates off the exact manifold; degeneracy can make the active structure nonunique; acceleration can converge rapidly to a nonglobal critical point.
Archive the manifold definition, tangent and normal spaces, subdifferential condition, nondegeneracy margin, residuals, detector settings, switch time, accepted and rejected accelerated steps and fallback behaviour. Report ambient and reduced convergence separately.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is five unchanged supports enough?
It is a diagnostic choice, not a theorem. The required guarantee comes from the stated identification result and its assumptions.
Why keep the proximal step after switching?
It protects descent and restores the global method when a local Newton model is premature or inaccurate.
Does superlinear local convergence start at identification?
Not necessarily. The iterate may need to enter a smaller neighbourhood where the second-order assumptions and step acceptance hold.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Hare and Lewis develop finite active-manifold identification under partial smoothness. Lewis and Wylie connect partial smoothness with active-set Newton methods. Bareilles, Iutzeler and Malick prove Newton acceleration on manifolds identified by proximal gradient. The ledgers above are original teaching examples.
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