To understand how high-dimensional inference becomes feasible, you must first grasp the concept of sparsity. In many high-dimensional problems, the underlying parameter or signal of interest is not arbitrary but instead possesses a special structure: it is sparse. A vector in $\mathbb{R}^p$ is called k-sparse if it has at most $k$ nonzero entries, where $k \ll p$. The set of indices corresponding to the nonzero entries is called the support set. For a vector $\beta$ in $\mathbb{R}^p$, the support set is defined as $\operatorname{supp}(\beta) = \{j : \beta_j \neq 0\}$. The effective dimensionality of a k-sparse vector is thus $k$, even though the ambient dimension $p$ may be much larger.

This notion of sparsity is central to modern high-dimensional statistics. Rather than attempting to estimate all $p$ parameters, you focus on the much smaller set of $k$ nonzero coefficients. This reduction in effective dimensionality is what enables statistical inference in situations where $p \gg n$ (the number of variables far exceeds the number of observations).

Sparse modeling is most commonly encountered in the context of sparse linear models. In the standard linear regression setup, you observe $n$ samples $(x_i, y_i)$ and posit a linear relationship $y = X\beta + \varepsilon$, where $X$ is the $n \times p$ design matrix, $\beta$ is a $p$-dimensional parameter vector, and $\varepsilon$ is noise. The key assumption in sparse models is that $\beta$ is $k$-sparse for some small $k$. This assumption is crucial for identifiability: if $\beta$ could be arbitrary, then with $p \gg n$ there would be infinitely many solutions for $\beta$ that fit the data perfectly. However, if you know that only $k$ coefficients are nonzero, and $k$ is small relative to $n$, it becomes possible to uniquely identify or approximate the true $\beta$.

Sparsity plays a vital role in overcoming the curse of dimensionality. In high-dimensional spaces, classical estimators fail because the number of parameters to estimate grows too quickly relative to the available data. By leveraging the assumption that the true model is sparse, you restrict attention to a vastly smaller subset of parameter space. This enables successful estimation, prediction, and even variable selection in high-dimensional regimes where traditional methods break down.