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- **Traditional Approach:** This matrix is often sparsified in practice by simply ignoring small elements (e.g., zeroing out elements less than 1% of the maximum value in $𝐴$), which can potentially lead to sub-optimal treatment plans.
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- **CompressRTP Solutions:** We provide a compressed and accurate representation of matrix $𝐴$ using two different techniques:
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- **(1.1) Sparse-Only Compression:** This technique sparsifies $𝐴$ using advanced tools from probability and randomized linear algebra. ([NeurIPS paper](./images/RMR_NeurIPS_Paper.pdf), [Sparse-Only Jupyter Notebook](https://github.com/PortPy-Project/CompressRTP/blob/main/examples/matrix_sparse_only.ipynb))
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- **(1.2) Sparse-Plus-Low-Rank Compression:** This method decomposes $𝐴$ into a sum of a sparse matrix and a low-rank matrix. ([ArXiv paper](https://arxiv.org/abs/2410.00756), [Sparse-Plus-Low-Rank Jupyter Notebook](https://github.com/PortPy-Project/CompressRTP/blob/main/examples/matrix_sparse_plus_low_rank.ipynb))
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- **(1.2) Sparse-Plus-Low-Rank Compression:** This method decomposes $𝐴$ into a sum of a sparse matrix and a low-rank matrix. ([Medical Physics paper](https://aapm.onlinelibrary.wiley.com/doi/full/10.1002/mp.17736), [Sparse-Plus-Low-Rank Jupyter Notebook](https://github.com/PortPy-Project/CompressRTP/blob/main/examples/matrix_sparse_plus_low_rank.ipynb))
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2. **Fluence Compression to Enforce Smoothness on $𝑥$:**
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- **The Need for Smoothness:** The beamlet intensities $𝑥$ need to be smooth for efficient and accurate delivery of radiation. Smoothness refers to small variations in the intensity of neighboring beamlets in two dimensions.
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- **Traditional Approach:** Smoothness is often achieved implicitly by adding regularization terms to the objective function that discourage variations between neighboring beamlets.

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