归一化数据下Lasso的复杂性：几何与组合学
Complexity of Lasso with Normalized Data, Geometry and Combinatorics
AI总结：
该研究将Lasso复杂性与极值组合学、代数图论关联，推导归一化数据下Lasso解的闭式公式，证明其线段相交p维卦限的数量上界小于非归一化情况的上界。
摘要：
我们通过将该问题与极值组合学和代数图论关联，观察并证明了归一化数据下Lasso的复杂性小于非归一化数据的情况。我们采用几何方法研究Lasso，将其视为最小二乘目标函数的水平集与ℓ₁范数等于t的ℝᵖ中参数的多面体边界集B(t)的相切问题。在满秩假设下，我们从几何上推导了Lasso解的闭式精确公式。我们确立了Lasso解的重要一般性质，已知Lasso解可表示为ℝᵖ中的简单折线。从p=2和p=3开始，我们展示了归一化数据与非归一化数据下，Lasso解的折线可相交的p维卦限的最大数量存在显著差异。我们证明，在归一化情况下，h_{p,2}是Lasso解的线段相交p维卦限数量的一般上界，其中h_{p,r}是长度为p的二进制字中每两个字至少匹配r个位置的最大数量，r≤p。我们利用谱图论证明，当p为偶数时，h_{p,2}=2^{p-1}-C(p,p/2)/2；当p为奇数时，h_{p,2}=2^{p-1}-C(p-1,(p-1)/2)。已知对于一般数据，该数量的精确估计为2^{p-1}，我们将其等同于h_{p,1}。我们还得到了p维归一化数据下Lasso解的总线段数的上界，该上界显著小于非归一化情况下已知的精确上界(3ᵖ+1)/2。
英文摘要：
We observe and prove that the complexity of lasso for normalized data is smaller than for nonnormalized ones by relating this question to extremal combinatorics and algebraic graph theory. We employ a geometric approach to the lasso as a study of the tangency of the level sets of the least square objective function with the polyhedral boundary sets $B(t)$ of the parameters in $\mathbb R^p$ with the $\ell_1$ norm equal to $t$. We geometrically derive closed exact formulae for the solution of the lasso under the full rank assumption. We establish important general properties of the solutions of the lasso, which are known to be represented as a simple polygonal chain in $\mathbb{R}^p$. Starting from $p=2$ and $p=3$, we show a striking difference in the maximal number of $p$-dimensional orthants a polygonal chain of a lasso solution can intersect in the case of normalized data vs. nonnormalized data. We prove that in the normalized case, the number $h_{p,2}$ is a general upper bound for the number of segments of a lasso solution intersecting a $p$-dimensional orthant, where $h_{p,r}$ is the maximal number of binary words of length $p$ such that every two words match at least at $r$ spots, $r\le p$. We prove, using spectral graph theory, that $h_{p,2}=2^{p-1}-\binom{p}{p/2}/2$ for $p$ even and $h_{p,2}=2^{p-1}-\binom{p-1}{(p-1)/2}$ for $p$ odd. It was known that for general data the sharp estimate for that number is $2^{p-1}$, which we identify with $h_{p,1}$. We also find an upper bound for the total number of segments of a lasso solution with normalized data in dimension $p$, e.g. $\hat N(2\ell)=\frac{3^{2\ell}+5}{2} - 4\ell\, {}_3F_2\!\left( \begin{matrix} \frac{1}{2},\;1-\ell,\;\frac{1}{2}-\ell\\[1mm] \frac{3}{2},\;1 \end{matrix} ;4 \right)$, that is significantly less than $(3^p+1)/2$, a well-known sharp upper bound for the nonnormalized case.
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