In the MT SGL two regularization parameters need
In the MT-SGL, two regularization parameters need to be specified: λ1 and λ2. Using recent results on norm regularization (Banerjee et al., 2014), it is possible to express both parameters via a single parameter as follows: and (Meier et al., 2008, Banerjee et al., 2014), where and are computed as:
The choices follow from the current understanding in the literature of the correct form these parameters, in particular, in terms of the dual norm of the gradient of the objective (Banerjee et al., 2014, Liu and Ye., 2010). Thus, the only parameter to be empirically chosen in MT-SGL is the scaling γ.
Python codes of the proposed algorithm are available at: https://bitbucket.org/XIAOLILIU/mtl-sgl.
Experimental results In this section, we present experimental results to demonstrate the effectiveness of the proposed MT-SGL on characterizing AD progression using a dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI) (Weiner et al., 2010).
Introduction “Sporadic” late-onset Alzheimer's disease (LOAD) is now considered to be a multifactorial disease with strong genetic determinants. The most robust and reproducible genetic risk factor, elucidated more than 2 decades ago, is the ε4 allele of the APOE gene that encodes apolipoprotein E (APOE) (Poirier et al., 1993, Strittmatter et al., 1993). APOE is an important cholesterol transporter in the brain, and its DMAP markedly increases after neuronal injury (Poirier et al., 1991). Many cholesterol metabolism–related genes such as BIN1, CLU, PICALM, ABCA7, ABCG1, and SORL1 have been recently classified among the top 20 LOAD susceptibility loci by some of the largest genome-wide association studies (GWASs) undertaken to date (Beecham et al., 2014, Lambert et al., 2013). Brains of patients with Alzheimer's disease (AD) are characterized by the presence of extracellular amyloid plaques and intracellular neurofibrillary tangles (NFTs). It has been suggested that neuronal cholesterol may affect the activity and production of the secretases involved in amyloid cleavage and deposition (Sun et al., 2015). Neuronal cholesterol may also play an important role in NFT formation by modulating tau phosphorylation via the isoprenoid cascade (Fan et al., 2001). As cholesterol does not cross the blood-brain barrier, its concentration in the brain is exclusively supplied by local synthesis and is independent from that in peripheral tissues (Zhang and Liu, 2015). Cholesterol is generated by de novo synthesis from acetyl-coenzyme A with the help of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a rate-limiting enzyme (Bjorkhem and Meaney, 2004). For proper transport, cholesterol combines with phospholipids and apolipoproteins E and J (APOE and CLU) and is secreted by ATP-binding cassette transporters such as ABCA1, ABCA7, and ABCG1 (Poirier et al., 2014). These secreted complexes termed high-density lipoproteins (HDLs) are then hydrolyzed by lipoprotein lipases (LPLs), which facilitates the subsequent uptake by neurons (Davies et al., 2012). After binding to cell surface receptors such as LDLR (low-density lipoprotein receptor), LRP1 (low-density lipoprotein receptor related protein 1), and SORL1, lipoprotein complexes undergo endocytosis with the help of PICALM and BIN1 gene products (Poirier et al., 2014). Free cholesterol is esterified by sterol O-acyltransferase (SOAT1) and can be stored in the form of lipid droplets. Spare cholesterol can be converted to 25-hydroxycholesterol (25-OH) by the enzyme CH25H or to 24S-hydroxycholesterol (24S-OH) by CYP46A1 and leaves the brain by crossing the blood-brain barrier (Brown and Jessup, 2009). Oxysterols such as 24S-OH and 25-OH are ligands for NR1H3 (alias LXR) and promote the transcriptional regulation of APOE and other cholesterol transporters (Wolozin, 2004). When oxysterols are in excess, they also inhibit sterol regulatory element binding transcription factor 2 (SREBF2), preventing further cholesterol synthesis (Spann and Glass, 2013). At the opposite, when cellular cholesterol levels are low, SREBF2 proteins are activated by cleavage and translocated from the Golgi apparatus to the nucleus. After entering the nucleus, SREBF2 acts as a transcription factor that binds to promoters of key genes involved in cholesterol synthesis and transport (Shimano, 2001).