# strange result from fused lasso estimator

Let us consider the following estimator:

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

which is called fused estimator, cf. PROPERTIES AND REFINEMENTS OF THE FUSED LASSO

Next, if I understand correctly, fusion after fusion of the signal $$y$$ does not change the result, i.e. if $$\hat{\alpha}^{F} = \underset{\alpha \in \mathbb{R}^{n}}{\arg \min} (\hat{\beta}^{F}_{i} - \alpha_{i})^{2} + \lambda_{1} \sum_{i=1}^{n-1}|\alpha_{i} - \alpha_{i+1}|,$$ then $$\hat{\alpha}^{F} = \hat{\beta}^{F}$$, because projection is idempotent.

Nevertheless, let us try

library("genlasso")
n=20
p0 = c(1/12,1/12,1/12,1/12,1/12,1/12,1/12,1/12,1/12,1/12,1/12,1/12)
lambda_f =  0.2
y = rmultinom(1, size = n, prob = p0)
a = fusedlasso1d(y, X = diag(1, length(p0), length(p0)), gamma = 0, approx = FALSE, maxsteps = 2000,
minlam = 0, rtol = 1e-07, btol = 1e-07, eps = 1e-4,
verbose = FALSE)
a_fl = softthresh(a, lambda =  lambda_f, gamma =  0)

b = fusedlasso1d(a_fl[,1], X = diag(1, length(p0), length(p0)), gamma = 0, approx = FALSE, maxsteps = 2000,
minlam = 0, rtol = 1e-07, btol = 1e-07, eps = 1e-4,
verbose = FALSE)
b_fl = softthresh(b, lambda =  lambda_f, gamma =  0)


One can check that $$a_{fl} \neq b_{fl}$$. I am confused here.

Define $$\begin{equation*} \hat{\beta}^{F}_C = \arg \min_{\beta \in \mathbb{R}^{n}} \|y-\beta\|_2^2 \text{ s.t. } \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| \leq C \end{equation*}$$ and $$\begin{equation*} \hat{\alpha}^{F}_C = \arg \min_{\alpha \in \mathbb{R}^{n}} \|\hat\beta^F_C-\alpha\|_2^2 \text{ s.t. } \sum_{i=1}^{n-1}|\alpha_{i} - \alpha_{i+1}| \leq C. \end{equation*}$$ Since projections are idempotent, it must be that $$\hat{\beta}^{F}_C=\hat{\alpha}^{F}_C$$ for all $$C$$.
By Lagrangian duality, we may equivalently write $$\begin{equation*} \hat{\beta}^{F}_C = \arg \min_{\beta \in \mathbb{R}^{n}} \|y-\beta\|_2^2 +\lambda(C, y) \sum_{i=1}^{n-1}|\beta_{i} - \beta_{i+1}| \end{equation*}$$ and $$\begin{equation*} \hat{\alpha}^{F}_C = \arg \min_{\alpha \in \mathbb{R}^{n}} \|\hat\beta^F_C-\alpha\|_2^2 + \lambda(C,\hat\beta^F) \sum_{i=1}^{n-1}|\alpha_{i} - \alpha_{i+1}| \end{equation*}$$ for some tuning parameter $$\lambda$$ that depends on $$C$$ and the data. Since $$y \neq \hat\beta^F$$, it doesn't follow that $$\lambda(C, y) = \lambda(C,\hat\beta^F)$$.