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A KL inequality is local to one critical point. An algorithmic tail may approach a whole compact cluster set before its destination is known. Uniformisation supplies one neighbourhood and one desingulariser that work around that set.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Frame the problem
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 · FRAME THE PROBLEM
1. Pointwise geometry is not yet a tail theorem
At each critical point x̄, the Kurdyka–Łojasiewicz property supplies a neighbourhood, a gap range and a desingularising function. Those objects may vary with x̄. An iterative sequence whose cluster points form a set could move among many such neighbourhoods, so one pointwise inequality cannot simply be applied to the entire tail.
Uniformisation bridges that gap under compactness and a common objective value on the cluster set. It produces shared constants and a common desingulariser in a tube around the set. The result is local around a set, not global over the whole domain.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. The hypotheses behind uniformisation
Let Ω be a compact set on which a proper lower-semicontinuous objective f is constant. Suppose f has the KL property at every point of Ω. A uniformised statement gives ε>0, η>0 and one concave desingulariser φ such that the KL inequality holds when a point lies within ε of Ω and its objective value is between f(Ω) and f(Ω)+η.
Compactness enables a finite subcover of pointwise neighbourhoods. Constancy aligns their reference level. Without these ingredients, the slope–gap relation may change without a shared bound along the sequence.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A finite-cover ledger
Imagine a compact critical arc covered by three local KL neighbourhoods. Each has its own admissible radius and gap ceiling. A proof selects a finite cover and shrinks to common safe values.
| Local chart | Radius εi | Gap ceiling ηi | Local role |
|---|---|---|---|
| U1 | 0.20 | 0.08 | Left arc |
| U2 | 0.12 | 0.05 | Middle overlap |
| U3 | 0.18 | 0.06 | Right arc |
| Uniform safe choice | ≤0.12 | ≤0.05 | Works after finite-cover construction |
The smallest radius and gap ceiling may look conservative, but they buy a statement that follows an unknown tail wherever it approaches the compact set.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. From descent data to entry into the uniform tube
A convergence proof first establishes that objective values decrease to f* and that the sequence is bounded. Its cluster set Ω is then nonempty and compact, and continuity along the sequence can show f=f* on Ω. Distance from iterates to Ω tends to zero.
| Proof task | Evidence | What it unlocks |
|---|---|---|
| Bounded iterates | Level-boundedness or direct estimate | Compact cluster set |
| Objective convergence | Monotone decrease and lower bound | Common limiting value |
| Asymptotic regularity | Step norms tend to zero | Cluster-set linkage |
| Distance to Ω | Subsequence argument | Eventual entry into uniform tube |
Once both distance and objective gap are below the uniform thresholds, every later iterate remains in the region where the same inequality can be used, provided the descent framework controls the tail.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Telescoping one inequality across the entire tail
Sufficient decrease relates an objective drop to the squared step. A relative-error condition relates subgradient size to recent steps. The uniform KL inequality converts the shared objective gap through φ. Concavity then makes differences of φ values pay for step lengths.
Summing the resulting inequality telescopes. The total remaining length is finite, so the iterates are Cauchy in a complete finite-dimensional setting and converge to one critical point. Uniformisation is the reason the same φ can be summed after the limit point is still unknown.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Critical sets do not imply motion along the set
A continuum of critical points can exist, but finite length prevents a tail from wandering indefinitely with nonzero travel. The sequence may approach the set before choosing a point; the proof then shows that its total remaining movement is summable.
This does not mean every method converges to the same point or that the selected point is a minimiser. Initialisation, update order and perturbations can select different critical points, including saddles when the algorithm allows them.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: matrix factorisation symmetries
Transfer the idea to a factorisation objective with rotational or scaling symmetries. Equivalent parameterisations can form a compact critical orbit after normalisation. Pointwise KL geometry around each representative is not enough until the tail is controlled near the orbit.
Check compactness carefully: an unnormalised scaling orbit may be noncompact. Quotient geometry, gauge fixing or coercive regularisation may be needed before a compact-set uniformisation applies. The symmetry cannot be ignored by calling the orbit one point.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connecting set uniformisation to existing topology guides
The KL exponent guide connects local power geometry to rate classes. The desingularising-functions guide keeps the full non-power modulus. Uniformisation explains when one such modulus can govern a whole algorithmic tail near a compact critical set.
Use Fejér monotonicity and convergence gaps for related tail-control ideas that do not automatically invoke KL geometry. The assumptions and conclusions should remain distinct.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Uniformisation does not hold around an arbitrary noncompact set, nor when f takes different values across the set without reformulation. It does not establish boundedness, sufficient decrease or relative error. Those are separate parts of the convergence argument.
A finite cover does not mean one may simply choose any local desingulariser and the smallest constants. The construction must preserve the KL inequality and concavity. In infinite-dimensional spaces, compactness and the Cauchy-to-strong-convergence step require additional care.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
State the space, objective, subdifferential, generated sequence and cluster set. Prove boundedness, objective convergence, criticality of cluster points, constancy of f on Ω and compactness. Record the neighbourhood and gap restrictions used for uniformisation.
Then verify sufficient decrease, relative error, continuity along convergent subsequences and any inertial memory terms. Distinguish the theorem’s finite-length conclusion from empirical flattening caused by numerical precision. Report whether the result concerns iterates, objective values or residuals.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not wait until the limit point is known?
The convergence proof is what must establish that a single limit exists. Uniformisation lets the argument proceed while only the cluster set is known.
Must the cluster set contain several points?
No. Uniformisation also works when it is a singleton, but the set formulation avoids assuming that conclusion in advance.
Does finite length give a rate?
It gives convergence and a tail-length bound. A quantitative rate needs more information about the desingulariser and the descent constants.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Bolte, Sabach and Teboulle use the uniformised KL property in the PALM convergence framework. Attouch, Bolte and Svaiter develop convergence of descent methods for tame problems. Wang and Wang analyse the exact modulus of the generalised concave KL property. The cover ledgers above are original teaching examples.
Continue through the Topology root, desingularising functions and custom rates and metric subregularity and error bounds. The Gold Standard of Education and How X Works provide the wider route.
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