Verteporfin

A 20 kb genomic fragment containing Il10 along with flanking sequences was isolated via recombineering from BAC clone RP23-122P5 (BACPAC Resource Center, Children’s Hospital Oakland). A 6.8 kb EcoRI fragment containing the 5^th exon, the endogenous polyA site and the 3′ UTR of Il10 was subsequently subcloned into pBluescript II KS (Stratagene). A floxed neomycin-IRES-eGFP cassette (15 (link)) was cloned into the HindIII site between the endogenous stop and polyA sites of il10, followed by subcloning of an HSV-TK cassette (16 (link)) into the SalI site of pBluescript II KS. After linearizing with NotI, the targeting vector was electroporated into a C57BL/6 ES cell line, and ES cell clones were selected with G418 and gancyclovir. Correctly targeted clones, screened initially by PCR followed by Southern blot confirmation (17 (link)) were injected into C57BL/6 albino blastocysts implanted into C57BL/6 females. Male chimeric mice were bred to C57BL/6 albino females to screen for germ line transmission. The neomycin cassette was floxed-out using C57BL/6 Zp3-Cre mice (Jackson laboratories, Bar Harbor), and correctly targeted heterozygous mice were interbred to generate homozygous Vert-X (Vert, fr. green; X, roman numeral 10) mice. Genotyping of Vert-X mice was performed by PCR using the following oligonucleotides: (a) il10 5′ ACCAAGGTGTCTACAAGGCCATGAATGAATT; (b) GFP 5′ GAGGAAATTGCATCGCATTGTCTGAGTAGGT; (c) il10 3′ CAAAGGCAGACAAACAATACACCATTCCCA.

We can build further on the connection between (2) and (5) discussed in Remark 3. Our removal of the symmetry constraint in (7) is analogous to the technique of duplicating columns of the design matrix used in the overlap group lasso [Obozinski, Jacob and Vert (2011) ].
A favorable property that distinguishes our method from previous approaches discussed in Section 4 is the relative transparency of the role that the hierarchy constraint plays in our estimator. This aspect is developed in Section 3.1.
Although our primary focus in this paper is on the Gaussian setting of (1), our proposal extends straightforwardly to other situations, such as the logistic regression setting in which the response is binary. In this case, we simply have q(β₀, β, Θ) be the appropriate negative log-likelihood,
$-{\sum }_{i=1}^n{y}_{i}log{p}_i+(1-y_i)log(1-p_i)$ , where
$p_i={[1+exp(-{\beta }_0-x_i^T\beta -\frac{1}{2}{x}_i^T\mathrm{\Theta }{x}_i)]}^{-1}$ . In Section 3 of the supplementary materials [Bien, Taylor and Tibshirani (2013) (link)], we show that solving this problem requires only a minor modification to our primary algorithm. It should also be noted that our estimator (and the algorithms developed to compute it) is designed for both the p < n and p ≥ n setting.
As a preliminary example, consider predicting whether a sample of olive oil comes from Southern Apulia based on measurements of the concentration of p = 8 fatty acids [Forina et al. (1983) ]. The dataset consists of n = 572 samples, and we average our results over 100 random equal-sized train-test splits. We compare three methods: (a) a standard lasso with main effects only (MEL), (b) the all-pairs lasso (APL), and (c) the strong hierarchical lasso (HL).
The top left panel of Figure 1 shows an interesting difference between HL and APL. We see that, on average, at a parameter sparsity level of five, the HL model uses four of the measured variables whereas APL uses six. Using the hierarchical model to classify a future olive oil, we only need to measure four rather than six of the fatty acids.
The top right panel of Figure 1 shows the predictive performance (versus the practical sparsity) of the three methods. It appears that HL enjoys the “best of both worlds,” matching the good performance of MEL for low practical sparsity levels (since it tends to pick out the main effects first) and the good performance of APL at high practical sparsity levels (since it can incorporate predictive interactions). Finally, the bottom panel of the figure provides a visual display of a sequence of HL’s solutions (by varying λ). Nonzero main effects are shown as filled nodes, and edges indicate nonzero interactions. Since all edges are incident to filled nodes, we see that strong hierarchy holds.
In the next section, we present several properties of our estimator that shed light on the effect of adding the convex hierarchy constraint to the lasso. Among these properties is an unbiased estimate of the degrees of freedom of our estimator. We view this degrees-of-freedom result as valuable primarily for the sake of understanding the effect of hierarchy. While such an estimate could be used for parameter selection, we prefer cross validation to select λ since this is more directly tied to the goal of prediction.