Non-negative sparse coding (NNSC) is a method for decomposing multivariate data into non-negative sparse components. This paper describes the motivation behind this approach and its relation to standard sparse coding and non-negative matrix factorization (NMF). A simple and efficient multiplicative algorithm is proposed for finding the optimal hidden components, and the basis vectors are learned from the data. Simulations demonstrate the effectiveness of the method. Linear data representations are widely used in signal processing and data analysis. Traditional methods like Fourier and wavelet analysis are mathematically sound but not data-adaptive. Data-adaptive representations, such as PCA, ICA, sparse coding, and NMF, are learned directly from the data. NNSC combines sparse coding and NMF, aiming to learn parts-based representations. It assumes that the data, basis vectors, and hidden components are non-negative. The goal of NNSC is to minimize an objective function that balances reconstruction error and sparsity. The objective function is: $$ C(\mathbf{A},\mathbf{S})=\frac{1}{2}\|\mathbf{X}-\mathbf{A}\mathbf{S}\|^{2}+\lambda\sum_{ij}S_{ij} $$ with constraints on the non-negativity of A and S, and the unit norm of the columns of A. The sparsity is measured by a linear activation penalty. An efficient algorithm is proposed for optimizing the hidden components S, given a fixed basis A. The algorithm uses a multiplicative update rule that ensures non-negativity of S. For optimizing both A and S, a projected gradient descent approach is used, with steps that ensure the constraints are satisfied. Experiments show that NNSC can learn parts-based non-negative representations, outperforming NMF in identifying all features. NMF struggles with overcomplete representations, while NNSC benefits from sparsity. Experiments with natural images confirm the importance of sparsity in learning non-negative representations. NNSC is related to linear sparse coding and NMF, and also to independent component analysis (ICA). It is shown that the objective function can be interpreted as the negative joint log-posterior in a noisy ICA model. The method is also related to recent work on non-negativity constraints in ICA. The paper concludes that NNSC is a useful method for learning parts-based representations from non-negative data, with a simple and efficient algorithm for estimating the hidden components.Non-negative sparse coding (NNSC) is a method for decomposing multivariate data into non-negative sparse components. This paper describes the motivation behind this approach and its relation to standard sparse coding and non-negative matrix factorization (NMF). A simple and efficient multiplicative algorithm is proposed for finding the optimal hidden components, and the basis vectors are learned from the data. Simulations demonstrate the effectiveness of the method. Linear data representations are widely used in signal processing and data analysis. Traditional methods like Fourier and wavelet analysis are mathematically sound but not data-adaptive. Data-adaptive representations, such as PCA, ICA, sparse coding, and NMF, are learned directly from the data. NNSC combines sparse coding and NMF, aiming to learn parts-based representations. It assumes that the data, basis vectors, and hidden components are non-negative. The goal of NNSC is to minimize an objective function that balances reconstruction error and sparsity. The objective function is: $$ C(\mathbf{A},\mathbf{S})=\frac{1}{2}\|\mathbf{X}-\mathbf{A}\mathbf{S}\|^{2}+\lambda\sum_{ij}S_{ij} $$ with constraints on the non-negativity of A and S, and the unit norm of the columns of A. The sparsity is measured by a linear activation penalty. An efficient algorithm is proposed for optimizing the hidden components S, given a fixed basis A. The algorithm uses a multiplicative update rule that ensures non-negativity of S. For optimizing both A and S, a projected gradient descent approach is used, with steps that ensure the constraints are satisfied. Experiments show that NNSC can learn parts-based non-negative representations, outperforming NMF in identifying all features. NMF struggles with overcomplete representations, while NNSC benefits from sparsity. Experiments with natural images confirm the importance of sparsity in learning non-negative representations. NNSC is related to linear sparse coding and NMF, and also to independent component analysis (ICA). It is shown that the objective function can be interpreted as the negative joint log-posterior in a noisy ICA model. The method is also related to recent work on non-negativity constraints in ICA. The paper concludes that NNSC is a useful method for learning parts-based representations from non-negative data, with a simple and efficient algorithm for estimating the hidden components.