Tensor multiplication with learned weight matrices is the fundamental building block in deep learning models. These matrices can often be sparsified, decomposed, quantized, or subjected to random parameter sharing without losing accuracy, suggesting the possibility of more efficient transforms. Although many variants of weight matrices exist, unstructured ones are incompatible with modern hardware, slowing inference and training. On the other hand, structured variants often limit expressivity or fail to deliver the promised latency benefits. We present Sketch Structured Transform (SS1), an expressive and GPU-friendly operator that accelerates inference. SS1 leverages parameter sharing in a random yet structured manner to reduce computation while retraining the rich expressive nature of parameter sharing. We confirm empirically that SS1 offers better quality-efficiency tradeoffs than competing variants. Interestingly SS1 can be combined with Quantization to achieve gains unattainable by either method alone, a finding we justify via theoretical analysis. The analysis may be of independent interest.Moreover, existing pre-trained models can be projected onto SS1 and finetuned for efficient deployment. Surprisingly, these projected models can perform reasonably well even without finetuning. Our experiments highlight various applications of the SS1:(a) Training GPT2 and DLRM models from scratch for faster inference. (b) Finetuning projected BERT models for 1.31× faster inference while maintaining GLUE scores. (c) Proof of concept with Llama-3-8b, showing 1.11× faster wall clock inference using projected SS1 layers without finetuning. We open source our code :https://github.com/apd10/Sketch-Structured-Linear/