Tessellated Distributed Computing
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913324174671872 |
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| author | Khalesi, Ali Elia, Petros |
| author_facet | Khalesi, Ali Elia, Petros |
| contents | The work considers the $N$-server distributed computing scenario with $K$ users requesting functions that are linearly-decomposable over an arbitrary basis of $L$ real (potentially non-linear) subfunctions. In our problem, the aim is for each user to receive their function outputs, allowing for reduced reconstruction error (distortion) $ε$, reduced computing cost ($γ$; the fraction of subfunctions each server must compute), and reduced communication cost ($δ$; the fraction of users each server must connect to). For any given set of $K$ requested functions -- which is here represented by a coefficient matrix $\mathbf {F} \in \mathbb{R}^{K \times L}$ -- our problem is made equivalent to the open problem of sparse matrix factorization that seeks -- for a given parameter $T$, representing the number of shots for each server -- to minimize the reconstruction distortion $\frac{1}{KL}\|\mathbf {F} - \mathbf{D}\mathbf{E}\|^2_{F}$ overall $δ$-sparse and $γ$-sparse matrices $\mathbf{D}\in \mathbb{R}^{K \times NT}$ and $\mathbf{E} \in \mathbb{R}^{NT \times L}$. With these matrices respectively defining which servers compute each subfunction, and which users connect to each server, we here design our $\mathbf{D},\mathbf{E}$ by designing tessellated-based and SVD-based fixed support matrix factorization methods that first split $\mathbf{F}$ into properly sized and carefully positioned submatrices, which we then approximate and then decompose into properly designed submatrices of $\mathbf{D}$ and $\mathbf{E}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14203 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tessellated Distributed Computing Khalesi, Ali Elia, Petros Information Theory The work considers the $N$-server distributed computing scenario with $K$ users requesting functions that are linearly-decomposable over an arbitrary basis of $L$ real (potentially non-linear) subfunctions. In our problem, the aim is for each user to receive their function outputs, allowing for reduced reconstruction error (distortion) $ε$, reduced computing cost ($γ$; the fraction of subfunctions each server must compute), and reduced communication cost ($δ$; the fraction of users each server must connect to). For any given set of $K$ requested functions -- which is here represented by a coefficient matrix $\mathbf {F} \in \mathbb{R}^{K \times L}$ -- our problem is made equivalent to the open problem of sparse matrix factorization that seeks -- for a given parameter $T$, representing the number of shots for each server -- to minimize the reconstruction distortion $\frac{1}{KL}\|\mathbf {F} - \mathbf{D}\mathbf{E}\|^2_{F}$ overall $δ$-sparse and $γ$-sparse matrices $\mathbf{D}\in \mathbb{R}^{K \times NT}$ and $\mathbf{E} \in \mathbb{R}^{NT \times L}$. With these matrices respectively defining which servers compute each subfunction, and which users connect to each server, we here design our $\mathbf{D},\mathbf{E}$ by designing tessellated-based and SVD-based fixed support matrix factorization methods that first split $\mathbf{F}$ into properly sized and carefully positioned submatrices, which we then approximate and then decompose into properly designed submatrices of $\mathbf{D}$ and $\mathbf{E}$. |
| title | Tessellated Distributed Computing |
| topic | Information Theory |
| url | https://arxiv.org/abs/2404.14203 |