## Results

Examples provided are the standard Mnist-digits and Mnist-fashion.

###  Coreset Construction

The coreset points, with their weights attached are clustered by our simplistic kmedoid algorithm.  
The cost of clustering with these centers is then compared with the cost of clustering the whole original data obtained
with the crate [kmedoids](https://crates.io/crates/kmedoids) using the parallel [par_fastermap](https://docs.rs/kmedoids/0.5.0/kmedoids/fn.par_fasterpam.html).

The computation times, in seconds, given are system time elapsed and total cpu times (to account for parallelism) 


#### Results for coreset construction + basic weighted medoid  (L1 distance) 

The size of the coreset was set 0.11 * the number of data points. We asked 10 clusters.

We give the  cost of clustering the coreset and the cost (of dispatching a posteriori the whole original data to the medoids position obtained via coreset clustering.  

As the results are random we give  they are given in the form (mean +-standard deviation) obtained on a sample of 20 computations.  



#### Reference results for medoid computations (L1 distance) with par_fastermap

The timings takes into account the computing of the distance matrix (fully multithreaded).  
|  mnist  |    cost    | time(sys) s | time(cpu) s |
| :-----: | :--------: | :---------: | :---------: |
| digits  | 1.789 10^6 |     55      |    1660     |
| fashion | 2.181 10^6 |     53      |    1460     |



#### Results with 15 iterations in Kmedoids.


|  mnist  |    cost (coreset)     |   cost (whole data)   | time(sys) s | time(cpu) s |
| :-----: | :-------------------: | :-------------------: | :---------: | :---------: |
| digits  | (1.864 +- 0.025) 10^6 | (1.873 +- 0.022) 10^6 |      1      |     14      |
| fashion | (2.250 +- 0.041) 10^6 | (2.255 +- 0.043) 10^6 |      1      |     14      |

The results are, on the average at less than 5% above the reference cost obtained by par_fastermap, and consistently within 8% at 2 std deviations.


#### Results with 25 iterations in Kmedoids.


|  mnist  |    cost (coreset)     |   cost (whole data)   | time(sys) s | time(cpu) s |
| :-----: | :-------------------: | :-------------------: | :---------: | :---------: |
| digits  | (1.868 +- 0.021) 10^6 | (1.876 +- 0.021) 10^6 |     1.1     |     16      |
| fashion | (2.230 +- 0.026) 10^6 | (2.239 +- 0.022) 10^6 |     1.1     |     16      |

The results are slightly better in the fashion case with 2.6% of overestimation of the cost compared with par_fastermap. 
The results are, on the average less than 5% above the reference cost obtained by par_fastermap, and within 8% at 3 std deviations.

##### Results on the [$\alpha$,$\beta$] approximation can be found [here](./bmor.md)