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add cost estimates
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lorenzoschena committed Jul 25, 2024
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Expand Up @@ -212,15 +212,11 @@ cost of the MODULO decompositions, with and without the memory-saving option.
Given a dataset of size ``n_s \times n_t``, we consider the computation of ``n_r`` modes. When using the memory saving option, we refer to
``n_t' = n_t / n_p`` as the number of time steps in each partition, and to ``n_s' = n_s / n_p`` as the number of spatial points in each partition.

| | Phase 1: \(D\) | Phase 2: \(K\) | Phase 3: \(\Psi\) | Phase 4: \(\Phi\) |
|------------------|---------------------------------------|--------------------------------------------------|----------------------------------------------|-----------------------------------------------------------------|
| No Memory Saving | \(n_s \times n_t\) | \(n_t^2\) | \(n_t^2 + n_t \times n_r\) | \(n_s \times n_t + n_t \times n_r + n_s \times n_r\) |
| Memory Saving | / | \(n_s \times n_t' + n_t' \times n_t'\) | \(n_t^2 + r \cdot n_t\) | \((n_s \times n_t') + n_s' \times n_t + n_s \times n_r\) |


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| | Phase 1: $D$ | Phase 2: $K$ | Phase 3: $\\Psi$ | Phase 4: $\\Phi$ |
|------------------|------------------|--------------------------------------|--------------------------|------------------------------------------------------|
| No Memory Saving | $n_s \times n_t$ | $n_t^2$ | $n_t^2 + n_t \times n_r$ | $n_s \times n_t + n_t \times n_r + n_s \times n_r$ |
| Memory Saving | / | $n_s \times n_t' + n_t' \times n_t'$ | $n_t^2 + r \cdot n_t$ | $n_s \times n_t' + n_s' \times n_t + n_s \times n_r$ |
As shown in the table, the computational cost of the decomposition is dominated by the computation of the correlation matrix ``K`` in Phase 2. The
memory-saving option significantly reduces the computational cost of this phase, as the correlation matrix is computed for each partition, rather than
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