We introduce a new family of algorithms for detecting and estimating a rank-one signal from a noisy observation under prior information about that signal's direction, focusing on examples where the signal is known to have entries biased to be positive. Given a matrix observation 𝐘, our algorithms construct a nonlinear Laplacian, another matrix of the form 𝐘 + diag(σ(𝐘1)) for a nonlinear σ: ℝ→ℝ, and examine the top eigenvalue and eigenvector of this matrix. When 𝐘 is the (suitably normalized) adjacency matrix of a graph, our approach gives a class of algorithms that search for unusually dense subgraphs by computing a spectrum of the graph "deformed" by the degree profile 𝐘1. We study the performance of such algorithms compared to direct spectral algorithms (the case σ = 0) on models of sparse principal component analysis with biased signals, including the Gaussian planted submatrix problem. For such models, we rigorously characterize the critical threshold strength of rank-one signal, as a function of the nonlinearity σ, at which an outlier eigenvalue appears in the spectrum of a nonlinear Laplacian. While identifying the σ that minimizes this critical signal strength in closed form seems intractable, we explore three approaches to design σ numerically: exhaustively searching over simple classes of σ, learning σ from datasets of problem instances, and tuning σ using black-box optimization of the critical signal strength. We find both theoretically and empirically that, if σ is chosen appropriately, then nonlinear Laplacian spectral algorithms substantially outperform direct spectral algorithms, while avoiding the complexity of broader classes of algorithms like approximate message passing or general first order methods.