ORLS-SVDD Lab

Sequential frozen-weight ORLS-SVDD

From un to α, iteration by iteration

Historical weights stay frozen while each arriving observation is optimized to convergence.

Dual coefficients

Recompute all dual coefficients from the current inverse; their entries always sum to one.

u3 → α iteration ledger

u₁, u₂ and their C values remain fixed.

ku3(k)C3(k)H33(k)s3(k)α(k)z3(k)u3(k+1)
0-1.0000001.0000001.5000001.330144(0.372340, 0.276596, 0.351064)0.017730-0.993732
1-0.9937320.9937321.5031541.333297(0.372576, 0.277007, 0.350417)0.018557-0.993136
2-0.9931360.9931361.5034561.333599(0.372598, 0.277047, 0.350355)0.018636-0.993078
3-0.9930780.9930781.5034851.333629(0.372600, 0.277050, 0.350349)0.018644-0.993072
4-0.9930720.9930721.5034881.333632(0.372600, 0.277051, 0.350349)0.018645-0.993071✓
Select any row to inspect its complete state above.5 iterations · final |Δu| 5.63 × 10⁻7

Selected state

What changed when x3 arrived?

Current values follow the selected ledger row. “Before arrival” means the converged state after x2. “Previous iteration” means inner step k−1.

Current systemH3(4)

Only the newest diagonal uses C3(4) = 0.993072. The historical block is frozen.

Current inverseH3−1

This inverse produces the selected row’s complete α vector.

Coefficient historyα(4)
Current[0.372600, 0.277051, 0.350349]ᵀ
Previous inner iteration[0.372600, 0.277050, 0.350349]ᵀ
Before x3 arrived[0.500000, 0.500000]ᵀ
Before the arrival

Stored state after x2

This is the frozen upper-left block carried into the x3 calculation.

H2
H2−1
Compare H with the previous inner iteration
Iteration 3
Newest diagonal change0.000003

H33(4) − H33(3)

Restricted HQ optimization

What happens inside one arrival step

The superscript k is the inner iteration. The subscript n identifies the newly arriving observation.

01

Initialize the new weight

un(0) = −1

Earlier auxiliary variables are already part of the stored online state.

02

Convert weight to regularization

Cn(k) = Cβν(−un(k))

Here Cβν = 1, so the numerical value of Cn is simply −un.

03

Border the frozen system

hnn(k) = k(xn,xn) + 1/(2Cn(k))

The upper-left historical block Hn−1 is unchanged by design.

04

Update the inverse

sn(k) = hnn(k) − δn⊤an

With an = Hn−1−1δn, this scalar Schur complement determines the full block inverse.

05

Recompute all dual coefficients

α(k) = ½(Hn(k))−1[kn + η(k)e]

Although only the new robust weight changes, every component of α responds to the new inverse. The equality constraint remains e⊤α = 1.

Frozen during the inner loop

Historical robust state

  • u₁,…,un−1
  • C₁,…,Cn−1
  • The upper-left block Hn−1
Updated during the inner loop

New-point coupled state

  • un(k) and Cn(k)
  • Hn(k) and its inverse
  • All αi(k), R², and zn(k)

Converged state

The final matrices after adding x₄

The three-point matrix is retained as the exact upper-left block. Only the fourth row, column, and adaptive diagonal are introduced.

H₄
H₄⁻¹
Compare with the frozen three-point state
H₃
H₃⁻¹

Numerical verification

The recursion returns the same sequential state

All comparisons use unrounded values. Differences at approximately 10⁻¹⁶ are ordinary floating-point roundoff.

ADD x₄1.11 × 10⁻¹⁶

Maximum H₄⁻¹ difference

Online block addition versus a four-point fit built with the identical frozen-history sequence.

Δu = 0ΔC = 0ΔH = 0
REMOVE x₄1.11 × 10⁻¹⁶

Original H₃⁻¹ restored

Adding and then removing the newest point returns the previous sequential state.

Δu = 0ΔC = 0ΔH = 0
REMOVE x₁1.11 × 10⁻¹⁶

Retained-state inverse

The downdate agrees with direct inversion on x₂,x₃,x₄ using their stored historical weights.

u retainedC retainedH exact