Nd-structure generation and operations

TODO

Performance for n-dimensional structures operations

One of the most sought after features of mathematical libraries is the high-performance operations on n-dimensional structures. In kmath performance depends on which particular context was used for operation.

Let us consider following contexts:

    // automatically build context most suited for given type.
    val autoField = NDField.auto(RealField, dim, dim)
    // specialized nd-field for Double. It works as generic Double field as well
    val specializedField = NDField.real(dim, dim)
    //A generic boxing field. It should be used for objects, not primitives.
    val genericField = NDField.buffered(intArrayOf(dim, dim), RealField)

Now let us perform several tests and see which implementation is best suited for each case:

Test case

In order to test performance we will take 2d-structures with dim = 1000 and add a structure filled with 1.0 to it n = 1000 times.

Specialized

The code to run this looks like:

    specializedField.run {
        var res: NDBuffer<Double> = one
        repeat(n) {
            res += 1.0
        }
    }

The performance of this code is the best of all tests since it inlines all operations and is specialized for operation with doubles. We will measure everything else relative to this one, so time for this test will be 1x (real time on my computer is about 4.5 seconds). The only problem with this approach is that it requires to specify type from the beginning. Everyone do so anyway, so it is the recommended approach.

Automatic

Let's do the same with automatic field inference:

    autoField.run {
        var res = one
        repeat(n) {
            res += 1.0
        }
    }

Ths speed of this operation is approximately the same as for specialized case since NDField.auto just returns the same RealNDField in this case. Of course it is usually better to use specialized method to be sure.

Lazy

Lazy field does not produce a structure when asked, instead it generates an empty structure and fills it on-demand using coroutines to parallelize computations. When one calls

    lazyField.run {
        var res = one
        repeat(n) {
            res += 1.0
        }
    }

The result will be calculated almost immediately but the result will be empty. In order to get the full result structure one needs to call all its elements. In this case computation overhead will be huge. So this field never should be used if one expects to use the full result structure. Though if one wants only small fraction, it could save a lot of time.

This field still could be used with reasonable performance if call code is changed:

    lazyField.run {
        val res = one.map {
            var c = 0.0
            repeat(n) {
                c += 1.0
            }
            c
        }

        res.elements().forEach { it.second }
    }

In this case it completes in about 4x-5x time due to boxing.

Boxing

The boxing field produced by

    genericField.run {
        var res: NDBuffer<Double> = one
        repeat(n) {
            res += 1.0
        }
    }

obviously is the slowest one, because it requires to box and unbox the double on each operation. It takes about 15x time (TODO: there seems to be a problem here, it should be slow, but not that slow). This field should never be used for primitives.

Element operation

Let us also check the speed for direct operations on elements:

    var res = genericField.one
    repeat(n) {
        res += 1.0
    }

One would expect to be at least as slow as field operation, but in fact, this one takes only 2x time to complete. It happens, because in this particular case it does not use actual NDField but instead calculated directly via extension function.

What about python?

Usually it is bad idea to compare the direct numerical operation performance in different languages, but it hard to work completely without frame of reference. In this case, simple numpy code:

res = np.ones((1000,1000))
for i in range(1000):
    res = res + 1.0

gives the completion time of about 1.1x, which means that specialized kotlin code in fact is working faster (I think it is because better memory management). Of course if one writes res += 1.0, the performance will be different, but it would be differenc case, because numpy overrides += with in-place operations. In-place operations are available in kmath with MutableNDStructure but there is no field for it (one can still work with mapping functions).

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