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Near a degenerate boundary, an active label can flip back and forth even while the objective improves. Hysteresis and cooldown windows can keep the solver from switching modes on every small perturbation, provided they are treated as safeguards rather than proofs of identification.
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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. Chattering is a state-machine failure
An active-set detector often maps a continuous residual or margin into a discrete label. When the signal hovers near one threshold, tiny numerical or stochastic changes can alternate the label at every iteration.
Each flip can reset memory, rebuild a tangent model or trigger an expensive factorisation. The issue is not only visual noise: rapid switching can erase the computational gain that manifold identification was meant to provide.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Hysteresis separates entry from exit
A hysteretic detector uses a stronger condition to enter a manifold state and a weaker condition to remain there. For an inactive coordinate, entry might require a margin above 0.020 for several steps, while exit occurs only when the margin falls below 0.005 or another acceptance gate fails.
The gap prevents small movements inside the band from changing the state. The thresholds are algorithmic choices that must be scaled to residual accuracy and problem geometry; they do not establish a theorem by themselves.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A chattering trace
Consider a candidate manifold monitored by a signed identification margin. Positive values favour the current label.
| Iteration | Margin | Single-threshold label at 0.010 | Hysteretic state |
|---|---|---|---|
| 21 | 0.012 | Active | Candidate |
| 22 | 0.008 | Inactive | Candidate |
| 23 | 0.011 | Active | Candidate |
| 24 | 0.009 | Inactive | Candidate |
| 25 | 0.023 | Active | Candidate 1 of 3 |
| 26 | 0.025 | Active | Candidate 2 of 3 |
| 27 | 0.024 | Active | Certified active |
Iterations 21–24 remain unresolved instead of repeatedly committing and undoing expensive local structure.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked entry, exit and cooldown rule
Use this illustrative rule: enter after three consecutive margins above 0.020; remain while the margin stays above 0.005 and local acceptance passes; exit immediately on a failed gate; then wait five accepted global steps before another entry attempt.
| Event | Observed condition | Rule action | Earliest new attempt |
|---|---|---|---|
| Entry | 0.023, 0.025, 0.024 | Certify at iteration 27 | Not applicable |
| Instability | Margin 0.003 at iteration 30 | Exit and reset memory | After five accepted global steps |
| Cooldown | Iterations 31–35 accepted | No re-entry tests | Iteration 36 |
| New candidate | Margins above 0.020 at 36–38 | Re-enter at 38 | Certified if other gates pass |
A failed objective or feasibility gate can override the margin because the manifold model must remain subordinate to global safeguards.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Choose windows from the error scale
Entry margins should exceed the combined scale of numerical error, stochastic noise and residual-estimation error. An arbitrary decimal threshold can be meaningless after rescaling variables or changing the penalty parameter.
Use relative or normalised margins where possible, and test nearby choices. If small changes reverse the reported acceleration benefit, the identification layer is not robust enough for a strong claim.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Reset the right state on exit
When the manifold changes, discard tangent-space curvature information that no longer has compatible coordinates. Reset quasi-Newton pairs, reduced factorizations and extrapolation states whose assumptions depend on the old label.
Keep global quantities that remain valid, such as a certified lower bound or monotone objective record. A restart should be surgical: remove geometry-specific memory without erasing evidence needed for convergence monitoring.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: rank identification
Transfer the rule to a low-rank matrix problem where a singular value hovers near the rank threshold. Enter a fixed-rank manifold only after a spectral gap persists; exit when the gap collapses or a local step fails.
The transfer check uses scale-aware singular-value margins, tangent consistency and actual descent. Hysteresis can reduce rank chatter, but it cannot make a truly multiple or zero spectral gap nondegenerate.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the active-geometry branch
The degenerate-manifold guide explains why identification can fail and when to restart. The identification-time guide separates detection from certification.
Use stratified critical sets when the correct geometry genuinely changes across pieces rather than merely chattering around one boundary.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Hysteresis can delay a legitimate switch as well as suppress false ones. A wide band improves stability but may keep the method on an obsolete manifold. Cooldown can waste iterations if the geometry changes quickly for real reasons.
Chattering-free means the implemented state machine avoids rapid label flips under its rules; it does not mean the continuous optimisation trajectory has identified the true active manifold. Keep the language distinct.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive raw margins, normalisation, entry and exit thresholds, persistence length, cooldown definition, gate failures, state transitions, memory resets, accepted fallback steps and wall-clock costs. Plot the signal beside the discrete state.
Test nearby thresholds, different noise levels and deliberately degenerate problems. Verify that the base method still progresses during cooldown and that rejected local trials cannot increase the protected objective or violate feasibility.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why use two thresholds?
The gap keeps small movements near a boundary from reversing the state on every iteration.
How long should cooldown last?
Long enough for the base method to rebuild reliable evidence, scaled to the problem and observed noise. It is an algorithmic design choice to validate.
Does persistence certify the manifold?
No. Persistence is evidence for a detector. Finite-identification theorems require convergence, partial smoothness and nondegeneracy conditions.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Hare and Lewis develop active-manifold identification through partial smoothness. Liang, Fadili and Peyré establish finite activity identification and local convergence for forward–backward methods. Sun, Jeong, Nutini and Schmidt give finite-detection conditions and degeneracy-sensitive bounds. The hysteresis and cooldown ledgers above are original algorithmic safeguards, not replacements for those theorems.
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