# alternative solution to fussed lasso

The question is related to strange result from fused lasso estimator

Let us consider fussed lasso estimator:

$$\hat{\beta}^{FL} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda_{1} \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| + \lambda_{2}\sum_{i=1}^{n}|\beta_{i}|],$$

From PATHWISE COORDINATE OPTIMIZATION it follows that the solution is given by the following: let $$\hat{\beta}^{F} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda_{1} \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}|],$$ then $$\hat{\beta}^{FL} = \underset{\alpha \in \mathbb{R}^{n}}{\arg \min} [(\hat{\beta}^{F}_{i} - \alpha_{i})^{2} + \lambda_{2}\sum_{i=1}^{n}|\alpha_{i}|],$$ i.e. first, one can perform fusion, and then we apply lasso to the fussed signal.

The question: is the following procedure correct?

Let us, first, solve the dual to Lagrange's form of fusion for some $$s_{1}>0$$:

$$\hat{\beta}^{*F} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2}] \, \text{ subject to } \, \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| \leq s_{1},$$ then, apply lasso to $$\hat{\beta}^{*F}$$: $$\hat{\xi} = \underset{\alpha \in \mathbb{R}^{n}}{\arg \min} [(\hat{\beta}^{*F}_{i} - \alpha_{i})^{2} + \lambda_{2}\sum_{i=1}^{n}|\alpha_{i}|].$$

Then, we can say that $$\hat{\xi} = \underset{\beta \in \mathbb{R}^{n}}{\arg \min} [(y_{i} - \beta_{i})^{2} + \lambda^{*}_{1}(y,s_{1}) \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| + \lambda_{2}\sum_{i=1}^{n}|\beta_{i}|],$$

i.e. there exists some $$\lambda^{*}_{1}$$, which depends on the signal and $$s_{1}$$.

• You can find a common algorithm to solve LASSO, smooth LASSO, fusion LASSO and elastic net in Nesterov's paper. You can also refer to Sara van de Geer's paper: doi.org/10.1214/11-EJS638. May 31 at 11:03
• the question is not about how to solve (there is R package available). I am more into theory and wonder if my conclusion is correct. May 31 at 11:06