Exact Sparsity

What Lasso Is and Is Not

The finale keeps the boundary visible: lambda is chosen, the data are tiny, and exact sparsity is one mechanical result, not a broad promise.

highlighted = computed this step

What is exact

The exact register contains the shown data, sum xy=28, sum x squared=14, the chosen λ values, and each rational weight.

∑xy=28,∑x2=14\sum xy=28,\quad \sum x^2=14
Lasso honesty boundaryExact sparsity and deferred claims.soft-threshold pathλmax(28-λ,0)w7213/2141412800λ=28 gives exact zeroridge versus lasso at λ=28penaltyformulaweightlasso L1max(28-λ,0)/140ridge L228/(14+λ)2/3lasso hits zero; ridge stays nonzerothrough-origin lasso fit; λ is chosen before the solve; L1 can hit exact zero; NOTconvergence; NOT general feature selection; NOT generalization

What is chosen

λ is chosen before the solve. The exact zero at λ=28 is a mechanical result of this objective and this data.

λ chosen;w(28)=0\lambda\text{ chosen};\quad w(28)=0
Lasso honesty boundaryExact sparsity and deferred claims.soft-threshold pathλmax(28-λ,0)w7213/2141412800λ=28 gives exact zeroridge versus lasso at λ=28penaltyformulaweightlasso L1max(28-λ,0)/140ridge L228/(14+λ)2/3lasso hits zero; ridge stays nonzerothrough-origin lasso fit; λ is chosen before the solve; L1 can hit exact zero; NOTconvergence; NOT general feature selection; NOT generalization

What lasso is and is not

This is one exact lasso solution at chosen λ values. It is NOT convergence, NOT general feature selection, NOT generalization, and NOT a claim about future data.

exact sparsity mechanics; deferred claims explicit\text{exact sparsity mechanics; deferred claims explicit}
Lasso honesty boundaryExact sparsity and deferred claims.soft-threshold pathλmax(28-λ,0)w7213/2141412800λ=28 gives exact zeroridge versus lasso at λ=28penaltyformulaweightlasso L1max(28-λ,0)/140ridge L228/(14+λ)2/3lasso hits zero; ridge stays nonzerothrough-origin lasso fit; λ is chosen before the solve; L1 can hit exact zero; NOTconvergence; NOT general feature selection; NOT generalization