  Generalized compressed sensing (GCS) is a paradigm in which a structured
high-dimensional signal may be recovered from random, under-determined, and
corrupted linear measurements. Generalized Lasso (GL) programs are effective
for solving GCS problems due to their proven ability to leverage underlying
signal structure. Three popular GL programs are equivalent in a sense and
sometimes used interchangeably. Tuned by a governing parameter, each admit an
optimal parameter choice. For sparse or low-rank signal structures, this choice
yields minimax order-optimal error. While GCS is well-studied, existing theory
for GL programs typically concerns this optimally tuned setting. However, the
optimal parameter value for a GL program depends on properties of the data, and
is typically unknown in practical settings. Performance in empirical problems
thus hinges on a program's parameter sensitivity: it is desirable that small
variation about the optimal parameter choice begets small variation about the
optimal risk. We examine the risk for these three programs and demonstrate that
their parameter sensitivity can differ for the same data. We prove a
gauge-constrained GL program admits asymptotic cusp-like behaviour of its risk
in the limiting low-noise regime. We prove that a residual-constrained Lasso
program has asymptotically suboptimal risk for very sparse vectors. These
results contrast observations about an unconstrained Lasso program, which is
relatively less sensitive to its parameter choice. We support the asymptotic
theory with numerical simulations, demonstrating that parameter sensitivity of
GL programs is readily observed for even modest dimensional parameters.
Importantly, these simulations demonstrate regimes in which a GL program
exhibits sensitivity to its parameter choice, though the other two do not. We
hope this work aids practitioners in selecting a GL program for their problem.
