Thanks a lot great explanation.
You have saved me a lot of work !
------
To clarify, it is *not* the case that `x.dot(spca.components_.T) ` is
equivalent to `spca.transform(x)`. The latter performs a solve.
Best,
Vlad
On Fri, Oct 17, 2014 at 12:03 PM, Vlad Niculae <zephyr14@...> wrote:
> Hi Luca
>
>> x_3_dimensional = x.dot(spca.components_.T) # this is equivalent to
>> spca.transform(x)
>
> This part is specific to PCA. In general, the transform part of such a
> decomposition is `X * components ^ -1`. In PCA, because `components`
> is orthogonal, `components ^ -1` is `components.T`. The relationship
> between SparsePCA and PCA omits the orthogonality constraints, but
> adds sparsity constraints instead. The connection between the is given
> (only) by the reconstruction objective, which for both is `min ||X -
> x_transf * components||`.
>
> What you get when you call `spca.transform` is essentially solving for
> x_transf in `min ||X - x_transf * components||` subject to the
> constraints. So the reconstruction is still `x_transf * components`.
>
> Hope this helps,
> Vlad
>
> On Fri, Oct 17, 2014 at 11:23 AM, Luca Puggini <lucapuggio@...> wrote:
>>
>> Hi Vlad thanks for the answer.
>>
>> I was thinking about that and I am not 100 % sure that this is right.
>>
>> If we consider SPCA to work as PCA than we do:
>>
>> x_3_dimensional = x.dot(spca.components_.T) # this is equivalent to
>> spca.transform(x)
>>
>> and so the reconstruction of x is x_reconstruction =
>> x_3_dimensional.dot(spca.components_)
>>
>> This works on PCA because the components are orthogonal
>> components_.dot(components_.T) = Id
>>
>> In SPCA this does not hold any more.
>>
>> So I am not sure that the reconstruction can be obtained in this way.
>>
>> What do you think about that?
>>
>> Thanks,
>>
>> Luca
>>
>>
>>
>> Hi Luca,
>>
>> The other part of the decomposition that you're missing is available
>> in `spca.components_` and has shape `(n_components, n_features)`. The
>> approximation of X is therefore `np.dot(x_3_dimensional,
>> spca.components_)`.
>>
>> Best,
>> Vlad
>>
>> On Thu, Oct 16, 2014 at 6:07 PM, Luca Puggini <lucapuggio@...> wrote:
>>> Hi,
>>> is there any way to reconstruct the data after SparsePCA?
>>>
>>> If I do
>>>
>>> spca = SparsePCA(alpha=1, n_components=3).fit(x)
>>> x_3_dimensional = SparsePCA.transform(x)
>>>
>>> How can I get the best lower rank approximation of x after SparsePCA
>>> decomposition?
>>>
>>> Thanks,
>>> Luca
>>
>>
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