Spectral clustering is a widely adopted method capable of identifying complicated cluster boundaries. However, traditional spectral clustering requires the definition of a predefined similarity metric for constructing the Laplacian matrix, a requirement that limits flexibility and adaptability. Instead of predefining this metric upfront as a fixed parametric function, we introduce a novel approach that learns the optimal parameters of a similarity function through parameter optimization. This optimizes a similarity function to assign high similarity values to data pairs with shared discriminative features and low values to those without such features. Previous methods that adapt similarity measures typically treat their parameters as hyperparameters or rely on non-convex optimization strategies. However, these approaches are not well-suited for unsupervised scenarios, as they depend heavily on initial conditions and require labeled data for validation, which is unavailable in such settings. In contrast, our method employs convex optimization to learn the parameters of the similarity metrics directly, rather than treating them as hyperparameters. This enables robust and reliable unsupervised learning, making our approach particularly well-suited for spectral clustering. We validate the effectiveness and adaptability of our method on several benchmark datasets, demonstrating superior performance compared to existing techniques.