I have been reading up on shadow mapping, and found the following tutorial:
http://www.opengl-tutorial.org/intermediate-tutorials/tutorial-16-shadow-mapping/
It makes sense to me up until the point where the author starts discussing the "shadow acne" artifact. They explain the cause with the following diagram (with no words):
I am still having a lot of trouble understanding what actually causes shadow acne and why adding a bias fixes it.
It seems that the resolution of the shadow map has no effect on acne. What is it then? Maybe float precision, or is it something else?
Yes, it is a precision issue. Not really a float problem, just finite precision.
In theory the shadow map stores "distance to closest object from light". But in practice it stores "distance±eps from light".
Then when testing, you have your fragments distance to the same light. But again, in practice ± eps2. So if you compare those two values it turns out that eps varies differently when interpolating for shadow map rendering or shading. So if you compare d ± eps < d2 ± eps2, if d2==d, you might get the wrong result because eps!=eps2. But if you compare d ± eps < d2 + max(eps) + max(eps2) ± eps2 you will be fine.
In this example d2==d. That is called self shadowing. And can be easily fixed with the above bias, or by simply not testing against yourself in raytracing.
It gets much more tricky with different objects and when eps and eps2 are vastly different. One way to deal with it is to control eps (http://developer.download.nvidia.com/SDK/10.5/opengl/src/cascaded_shadow_maps/doc/cascaded_shadow_maps.pdf). Or one can just take a lot more samples.
To try to answer the question: The core issue is that shadow mapping compares ideal distances. But those distances are not ideal but quantized. And quantized values are usually fine, but in this case we are comparing them in two different spaces.
Related
Hello stackoverflow community,
I have troubles in understanding a least-square-error-problem in the c++ armadillo package.
I have a matrix A with many more rows than columns (5000 to 100 for example) so it is overdetermined.
I want to find x so that A*x=b gives me the least square error.
If i use the solve function of armadillo on my data like "x = Solve(A,b)" the error of "(A*x-b)^2" is sometimes way to high.
If on the other hand I solve for x with the analytical form by "x = (A^T * A)^-1 *A^T * b" the results are always right.
The results for x in both cases can differ by 10 magnitudes.
I had thought that armadillo would use this analytical form in the background if the system is overdetermined.
Now I would like to understand why these two methods give such different results.
I wanted to give a short example program, but i can't reproduce this behavior with a short program.
I thought about giving the Matrix here, but with 5000 times 100 it's also very big. I can deliver the values for which this happens though if needed.
So as a short background.
The matrix I get from my program is a numerically solved reaction of a nonlinear oscillator in which I put information inside by wiggeling a parameter of this system.
Because the influence of this parameter on the system is small, the values of my different rows are very similar but never the same, otherwise armadillo should throw an error.
I'm still thinking that this is the problem, but the solve function never threw any error.
Another thing that confuses me is that in a short example program with a random matrix, the analytical form is waaay slower than the solve function.
But on my program, both are nearly identically fast.
I guess this has something to do with the numerical convergence of the pseudo inverse and the special case of my matrix, but for that i don't know enough about how armadillo works.
I hope someone can help me with that problem and thanks a lot in advance.
Thanks for the replies. I think i figured the problem out and wanted to give some feedback for everybody who runs into the same problem.
The Armadillo solve function gives me the x that minimizes (A*x-b)^2.
I looked at the values of x and they are sometimes in the magnitude of 10^13.
This comes from the fact that the rows of my matrix only slightly change. (So nearly linear dependent but not exactly).
Because of that i was in the numerical precision of my doubles and as a result my error sometimes jumped around.
If i use the rearranged analytical formular (A^T * A)*x = *A^T * b with the solve function this problem doesn't occur anymore because the fitted values of x are in the magnitude of 10^4. The least square error is a little bit higher but that is okay, as i want to avoid overfitting.
I now additionally added Tikhonov regularization by solving (A^T * A + lambda*Identity_Matrix)*x = *A^T * b with the solve function of armadillo.
Now the weight vectors are in the order of around 1 and the error nearly doesn't change compared to the formular without regularization.
When optimizing a SVGP with Poisson Likelihood for a big data set I see what I think are exploding gradients.
After a few epochs I see a spiky drop of the ELBO, which then very slowly recovers after getting rid of all progress made before.
Roughly 21 iterations correspond to an Epoch.
This spike (at least the second one) resulted in a complete shift of the parameters (for vectors of parameters I just plotted the norm to see changes):
How can I deal with that? My first approach would be to clip the gradient, but that seems to require digging around the gpflow code.
My Setup:
Training works via Natural Gradients for the variational parameters and ADAM for the rest, with a slowly (linearly) increasing schedule for the Natural Gradient Gamma.
The batch and inducing point sizes are as large as possible for my setup
(both 2^12, with the data set consisting of ~88k samples). I include 1e-5 jitter and initialize the inducing points with kmeans.
I use a combined kernel, consisting of a combination of RBF, Matern52, a periodic and a linear kernel on a total of 95 features (a lot of them due to a one-hot encoding), all learnable.
The lengthscales are transformed with gpflow.transforms.
with gpflow.defer_build():
k1 = Matern52(input_dim=len(kernel_idxs["coords"]), active_dims=kernel_idxs["coords"], ARD=False)
k2 = Periodic(input_dim=len(kernel_idxs["wday"]), active_dims=kernel_idxs["wday"])
k3 = Linear(input_dim=len(kernel_idxs["onehot"]), active_dims=kernel_idxs["onehot"], ARD=True)
k4 = RBF(input_dim=len(kernel_idxs["rest"]), active_dims=kernel_idxs["rest"], ARD=True)
#
k1.lengthscales.transform = gpflow.transforms.Exp()
k2.lengthscales.transform = gpflow.transforms.Exp()
k3.variance.transform = gpflow.transforms.Exp()
k4.lengthscales.transform = gpflow.transforms.Exp()
m = gpflow.models.SVGP(X, Y, k1 + k2 + k3 + k4, gpflow.likelihoods.Poisson(), Z,
mean_function=gpflow.mean_functions.Constant(c=np.ones(1)),
minibatch_size=MB_SIZE, name=NAME)
m.mean_function.set_trainable(False)
m.compile()
UPDATE: Using only ADAM
Following the suggestion by Mark, I switched to ADAM only,
which helped me get rid of that sudden explosion. However, I still initialized with an epoch of natgrad only, which seems to save a lot of time.
In addition, the variational parameters seem to change a lot less abrupt (in terms of their norm at least). I guess they'll converge way slower now, but at least it's stable.
Just to add to Mark's answer above, when using nat grads in non-conjugate models it can take a bit of tuning to get the best performance, and instability is potentially a problem. As Mark points out, the large steps that provide potentially faster convergence can also lead to the parameters ending up in in bad regions of the parameter space. When the variational approximation is good (i.e. the true and approximate posterior are close) then there is good reason to expect that the nat grad will perform well, but unfortunately there is no silver bullet in the general case. See https://arxiv.org/abs/1903.02984 for some intuition.
This is very interesting. Perhaps trying to not use natgrads is a good idea as well. Clipping gradients indeed seems like a hack that could work. And yes, this would require digging around in the GPflow code a bit. One tip that can help towards this, is by not using the GPflow optimisers directly. The model._likelihood_tensor contains the TF tensor that should be optimised. Perhaps with some manual TensorFlow magic, you can do the gradient clipping on here before running an optimiser.
In general, I think this sounds like you've stumbled on an actual research problem. Usually these large gradients have a good reason in the model, which can be addressed with careful thought. Is it variance in some monte carlo estimate? Is the objective function behaving badly?
Regarding why not using natural gradients helps. Natural gradients use the Fisher matrix as a preconditioner to perform second order optimisation. Doing so can result in quite aggressive moves in parameter space. In certain cases (when there are usable conjugacy relations) these aggressive moves can make optimisation much faster. This case, with the Poisson likelihood, is not one where there are conjugacy relations that will necessarily help optimisation. In fact, the Fisher preconditioner can often be detrimental, particularly when variational parameters are not near the optimum.
I am writing some simple Unit Tests for math library.
To decide if the library generates good results I have to compare them with expected ones. Because of rounding etc. even good result will differ a bit from expected one (e.g. 0.701 when 0.700 was expected).
The problem is, I have to decide how similar two vectors are. I want to describe that similarity as an error proportion (for number it would be e.g. errorScale(3.0f /* generated */, 1.0f /* expected */) = 3.0f/1.5f = 2.0f == 200%).
Which method should I use to determine the similarity of 2D, 3D and 4D (quaternions) vectors?
There's no universally good measure. In particular, for addition the absolute error is better while for multiplication the relative error is better.
For vectors the "relative error" can also be considered in terms of length and direction. If you think about it, the "acceptable outcomes" form a small area around the exact result. But what's the shape of this area? Is it an axis-aligned square (absolute errors in x and y direction)? That privileges a specific vector base. A circle might be a better shape.
I've seen this pseudo-random number generator for use in shaders referred to here and there around the web:
float rand(vec2 co){
return fract(sin(dot(co.xy ,vec2(12.9898,78.233))) * 43758.5453);
}
It's variously called "canonical", or "a one-liner I found on the web somewhere".
What's the origin of this function? Are the constant values as arbitrary as they seem or is there some art to their selection? Is there any discussion of the merits of this function?
EDIT: The oldest reference to this function that I've come across is this archive from Feb '08, the original page now being gone from the web. But there's no more discussion of it there than anywhere else.
Very interesting question!
I am trying to figure this out while typing the answer :)
First an easy way to play with it: http://www.wolframalpha.com/input/?i=plot%28+mod%28+sin%28x*12.9898+%2B+y*78.233%29+*+43758.5453%2C1%29x%3D0..2%2C+y%3D0..2%29
Then let's think about what we are trying to do here: For two input coordinates x,y we return a "random number". Now this is not a random number though. It's the same every time we input the same x,y. It's a hash function!
The first thing the function does is to go from 2d to 1d. That is not interesting in itself, but the numbers are chosen so they do not repeat typically. Also we have a floating point addition there. There will be a few more bits from y or x, but the numbers might just be chosen right so it does a mix.
Then we sample a black box sin() function. This will depend a lot on the implementation!
Lastly it amplifies the error in the sin() implementation by multiplying and taking the fraction.
I don't think this is a good hash function in the general case. The sin() is a black box, on the GPU, numerically. It should be possible to construct a much better one by taking almost any hash function and converting it. The hard part is to turn the typical integer operation used in cpu hashing into float (half or 32bit) or fixed point operations, but it should be possible.
Again, the real problem with this as a hash function is that sin() is a black box.
The origin is probably the paper: "On generating random numbers, with help of y= [(a+x)sin(bx)] mod 1", W.J.J. Rey, 22nd European Meeting of Statisticians and the 7th Vilnius Conference on Probability Theory and Mathematical Statistics, August 1998
EDIT: Since I can't find a copy of this paper and the "TestU01" reference may not be clear, here's the scheme as described in TestU01 in pseudo-C:
#define A1 ???
#define A2 ???
#define B1 pi*(sqrt(5.0)-1)/2
#define B2 ???
uint32_t n; // position in the stream
double next() {
double t = fract(A1 * sin(B1*n));
double u = fract((A2+t) * sin(B2*t));
n++;
return u;
}
where the only recommended constant value is the B1.
Notice that this is for a stream. Converting to a 1D hash 'n' becomes the integer grid. So my guess is that someone saw this and converted 't' into a simple function f(x,y). Using the original constants above that would yield:
float hash(vec2 co){
float t = 12.9898*co.x + 78.233*co.y;
return fract((A2+t) * sin(t)); // any B2 is folded into 't' computation
}
the constant values are arbitrary, especially that they are very large, and a couple of decimals away from prime numbers.
a modulus over 1 of a hi amplitude sinus multiplied by 4000 is a periodic function. it's like a window blind or a corrugated metal made very small because it's multiplied by 4000, and turned at an angle by the dot product.
as the function is 2-D, the dot product has the effect of turning the periodic function at an oblique relative to X and Y axis. By 13/79 ratio approximately. It is inefficient, you can actually achieve the same by doing sinus of (13x + 79y) this will also achieve the same thing I think with less maths..
If you find the period of the function in both X and Y, you can sample it so that it will look like a simple sine wave again.
Here is a picture of it zoomed in graph
I don't know the origin but it is similar to many others, if you used it in graphics at regular intervals it would tend to produce moire patterns and you could see it's eventually goes around again.
Maybe it's some non-recurrent chaotic mapping, then it could explain many things, but also can be just some arbitrary manipulation with large numbers.
EDIT: Basically, the function fract(sin(x) * 43758.5453) is a simple hash-like function, the sin(x) provides smooth sin interpolation between -1 to 1, so sin(x) * 43758.5453 will be interpolation from -43758.5453 to 43758.5453. This is a quite huge range, so even small step in x will provide large step in result and really large variation in fractional part. The "fract" is needed to get values in range -0.99... to 0.999... .
Now, when we have something like hash function we should create function for production hash from the vector. The simplest way is call "hash" separetly for x any y component of the input vector. But then, we will have some symmetrical values. So, we should get some value from the vector, the approach is find some random vector and find "dot" product to that vector, here we go: fract(sin(dot(co.xy ,vec2(12.9898,78.233))) * 43758.5453);
Also, according to the selected vector, its lenght should be long engough to have several peroids of the "sin" function after "dot" product will be computed.
I do not believe this to be the true origin, but OP's code is presented as code example in "The Book of Shaders" by Patricio Gonzalez Vivo and Jen Lowe ( https://thebookofshaders.com/10/ ). In their code, Patricio Gonzales Vivo is cited as the author, i.e "// Author #patriciogv - 2015"
Since the OP's research dates back even further (to '08), the source might at least explain its popularity, and the author might be able to shed some light on his source.
I develop a scientific application (simulation of chromosomes moving in a cell nucleus). The chromosomes are divided in small fragments that rotate around a random axis using 4x4 rotation matrices.
The problem is that the simulation performs hundreds of billions of rotations, therefore the floating-point rounding errors stack up and grow exponentially, so the fragments tend to "float away" and detach from the rest of the chromosome as time passes.
I use double precision with C++. The soft runs on CPU for the moment but will be ported for CUDA, and simulations can last for 1 month at most.
I have no idea how I could somehow renormalize the chromosome, because all fragments are chained together (you can see it as a doubly linked-list), but I think that would be the best idea, if possible.
Do you have any suggestions ? I feel a bit lost.
Thank you very much,
H.
EDIT:
Added a simplified sample code.
You can assume all matrix math are classical implementations.
// Rotate 1000000 times
for (int i = 0; i < 1000000; ++i)
{
// Pick a random section start
int istart = rand() % chromosome->length;
// Pick the end 20 segments further (cyclic)
int iend = (istart + 20) % chromosome->length;
// Build rotation axis
Vector4 axis = chromosome->segments[istart].position - chromosome->segments[iend].position;
axis.normalize();
// Build rotation matrix and translation vector
Matrix4 rotm(axis, rand() / float(RAND_MAX));
Vector4 oldpos = chromosome->segments[istart].position;
// Rotate each segment between istart and iend using rotm
for (int j = (istart + 1) % chromosome->length; j != iend; ++j, j %= chromosome->length)
{
chromosome->segments[j].position -= oldpos;
chromosome->segments[j].position.transform(rotm);
chromosome->segments[j].position += oldpos;
}
}
You need to find some constraint for your system and work to keep that within some reasonable bounds. I've done a bunch of molecular collision simulations and in those systems the total energy is conserved, so every step I double check the total energy of the system and if it varies by some threshold, then I know that my time step was poorly chosen (too big or too small) and I pick a new time step and rerun it. That way I can keep track of what's happening to the system in real time.
For this simulation, I don't know what conserved quantity you have, but if you have one, you can try to keep that constant. Remember, making your time step smaller doesn't always increase the accuracy, you need to optimize the step size with the amount of precision you have. I've had numerical simulations run for weeks of CPU time and conserved quantities were always within 1 part in 10^8, so it is possible, you just need to play around some.
Also, as Tomalak said, maybe try to always reference your system to the start time rather than to the previous step. So rather than always moving your chromosomes keep the chromosomes at their start place and store with them a transformation matrix that gets you to the current location. When you compute your new rotation, just modify the transformation matrix. It may seem silly, but sometimes this works well because the errors average out to 0.
For example, lets say I have a particle that sits at (x,y) and every step I calculate (dx, dy) and move the particle. The step-wise way would do this
t0 (x0,y0)
t1 (x0,y0) + (dx,dy) -> (x1, y1)
t2 (x1,y1) + (dx,dy) -> (x2, y2)
t3 (x2,y2) + (dx,dy) -> (x3, y3)
t4 (x3,30) + (dx,dy) -> (x4, y4)
...
If you always reference to t0, you could do this
t0 (x0, y0) (0, 0)
t1 (x0, y0) (0, 0) + (dx, dy) -> (x0, y0) (dx1, dy1)
t2 (x0, y0) (dx1, dy1) + (dx, dy) -> (x0, y0) (dx2, dy2)
t3 (x0, y0) (dx2, dy2) + (dx, dy) -> (x0, y0) (dx3, dy3)
So at any time, tn, to get your real position you have to do (x0, y0) + (dxn, dyn)
Now for simple translation like my example, you're probably not going to win very much. But for rotation, this can be a life saver. Just keep a matrix with the Euler angles associated with each chromosome and update that rather than the actual position of the chromosome. At least this way they won't float away.
Write your formulae so that the data for timestep T does not derive solely from the floating-point data in timestep T-1. Try to ensure that the production of floating-point errors is limited to a single timestep.
It's hard to say anything more specific here without a more specific problem to solve.
The problem description is rather vague, so here are some rather vague suggestions.
Option 1:
Find some set of constraints such that (1) they should always hold, (2) if they fail, but only just, it's easy to tweak the system so that they do, (3) if they do all hold then your simulation isn't going badly crazy, and (4) when the system starts to go crazy the constraints start failing but only slightly. For instance, perhaps the distance between adjacent bits of chromosome should be at most d, for some d, and if a few of the distances are just slightly greater than d then you can (e.g.) walk along the chromosome from one end, fixing up any distances that are too big by moving the next fragment towards its predecessor, along with all its successors. Or something.
Then check the constraints often enough to be sure that any violation will still be small when caught; and when you catch a violation, fix things up. (You should probably arrange that when you fix things up, you "more than satisfy" the constraints.)
If it's cheap to check the constraints all the time, then of course you can do that. (Doing so may also enable you to do the fixup more cheaply, e.g. if it means that any violations are always tiny.)
Option 2:
Find a new way of describing the state of the system that makes it impossible for the problem to arise. For instance, maybe (I doubt this) you can just store a rotation matrix for each adjacent pair of fragments, and force it always to be an orthogonal matrix, and then let the positions of the fragments be implicitly determined by those rotation matrices.
Option 3:
Instead of thinking of your constraints as constraints, supply some small "restoring forces" so that when something gets out of line it tends to get pulled back towards the way it should be. Take care that when nothing is wrong the restoring forces are zero or at least very negligible, so that they don't perturb your results more badly than the original numeric errors did.
I think it depends on the compiler you are using.
Visual Studio compiler support the /fp switch which tells the behavior of the floating point operations
you can read more about it. Basically, /fp:strict is the harshest mode
I guess it depends on the required precision, but you could use 'integer' based floating point numbers. With this approach, you use an integer and provide your own offset for the number of decimals.
For example, with a precision of 4 decimal points, you would have
float value -> int value
1.0000 -> 10000
1.0001 -> 10001
0.9999 -> 09999
You have to be careful when you do your multiply and divide and be careful when you apply your precision offsets. Other wise you can quickly get overflow errors.
1.0001 * 1.0001 becomes 10001 * 10001 / 10000
If I read this code correctly, at no time is the distance between any two adjacent chromosome segments supposed to change. In that case, before the main loop compute the distance between each pair of adjacent points, and after the main loop, move each point if necessary to have the proper distance from the previous point.
You may need to enforce this constraint several times during the main loop, depending on circumstances.
Basically, you need to avoid the accumulation of error from these (inexact) matrix operators and there are two major ways of doing so in most applications.
Instead of writing the position as some initial position operated on many times, you can write out what the operator would be explicitly after N operations. For instance, imagine you had a position x and you were adding a value e (that you couldn't represent exactly.) Much better than computing x += e; a large amount of times would be to compute x + EN; where EN is some more accurate way of representing what happens with the operation after N times. You should think whether you have some way of representing the action of many rotations more accurately.
Slightly more artificial is to take your newly found point and project off any discrepancies from the expected radius from your center of rotation. This will guarantee that it doesn't drift off (but won't necessarily guarantee that the rotation angle is accurate.)