2.2.1. *L*1/2-NMF

Sparsity is an inherent property of hyperspectral data. To reduce the solution space and derive results with expected sparsity levels, some sparseness regularizations are added to constrain the sparseness of abundances. The *L*1 regularizer is popular for generating sparse solutions. However, the *L*1 regularizer may not enforce a sufficiently sparse solution while preserving the additivity constraint over the abundances since the sum-to-one constraint is a fixed *L*1 norm. In [35], Qian et al. propose the *L*1/2-NMF, based on the *L*1/2 regularizer. The *L*1/2 regularizer possesses two advantages over the *L*1 regularizer. It can still enforce sparsity with the full additivity constraint imposed. Another advantage is that the *L*1/2 regularizer can obtain sparser solutions than the *L*1 regularizer does [36]. The model of NMF with the *L*1/2 regularizer is as follows: [35]

$$f(\mathcal{W}, H) = \frac{1}{2} \left\| X - \mathcal{W}H \right\|\_F^2 + \lambda \left\| H \right\|\_{1/2} \tag{5}$$

where

$$\|\|H\|\|\_{1/2} = \sum\_{p,n=1}^{P,N} H\_{pn}^{1/2} \tag{6}$$

and *Hpn* denotes the (*p*, *n*)-th element of *H*.

The objective in (5) is nonincreasing under the multiplicative update rules:

$$\mathcal{W} = \mathcal{W}.\*(\mathcal{X}H^T)./\mathcal{W}HH^T\tag{7}$$

$$H = H.\*(\mathcal{W}^T\mathcal{X})./(\mathcal{W}^T\mathcal{W}H + \frac{\lambda}{2}H^{-\frac{1}{2}})\tag{8}$$

where *H*− 12 denotes the reciprocal element-wise square root for each element in *H*.
