Sequencing of cell-free DNA (cfDNA) is currently being used to detect cancer by searching both for mutational and non-mutational alterations. Recent work has shown that the length distribution of cfDNA fragments from a cancer patient can inform tumor load and type. Here, we propose non-negative matrix factorization (NMF) of fragment length distributions as a novel and completely unsupervised method for studying fragment length patterns in cfDNA. Using shallow whole-genome sequencing (sWGS) of cfDNA from a cohort of patients with metastatic castration-resistant prostate cancer (mCRPC), we demonstrate how NMF accurately infers the true tumor fragment length distribution as an NMF component - and that the sample weights of this component correlate with ctDNA levels (r=0.75). We further demonstrate how using several NMF components enables accurate cancer detection on data from various early stage cancers (AUC = 0.96). Finally, we show that NMF, when applied across genomic regions, can be used to discover fragment length signatures associated with open chromatin.