9 ms·

I think Dimension in the data sense of the word, not Dimension in the spatial sense of the word. Multi-dimension data is a thing, and it goes a lot higher than

I think Dimension in the data sense of the word, not Dimension in the spatial sense of the word. Multi-dimension data is a thing, and it goes a lot higher than 4, 5, or 7.

gruez 5y ago
>I think Dimension in the data sense of the word Where else is that used? At least to me (and the common layperson), the only usage seems to be from people trying to hype up storage mediums. also, if we're using that definition, does that mean I have a 7D SSD? The NAND cells are arranged on both the planar axis, and stacked (3d NAND). That's 3 dimensions. Each NAND cell also encodes data using various charge levels, which currently tops out at 4 bits. In total that makes for 3 + 4 = 7 dimensions.
- kortex 5y ago
  That is still O(N^3) in my opinion. If you scale up the media in all directions, your data scales like the cubic. The charge level makes it maybe 3.1 dimensions, but it isn't really scalable. Now if you could add arbitrary voltage levels without losing precision, that would be 4D.
- emteycz 5y ago
  It's a math/information science term that was applied in physics (its most well known usage), not the other way around...
- lelandbatey 5y ago
  It's mostly a mathematical term which also applies to pieces of data. See here: https://en.wikipedia.org/wiki/Dimension https://en.wikipedia.org/wiki/Dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
- c5e3ebe93d2c 5y ago
  In my physics courses we would routinely talk about 6 dimensional spaces when discussing position-momentum space, also we would sometimes refer to two dimensional objects like a moebius strips as "1-dimensional". > also, if we're using that definition, does that mean I have a 7D SSD? Maybe. You can use whatever definitions you want if you feel it's helpful. However, based on your description, if you're using a single bit as a dimension, you should figure out how many bits the the dimensional information encodes. As it stands, it's not clear what you're counting as a 'dimension' in this context. Also, normally we think of degrees of freedom as being independent of each other (hence the claim to 5d in this example). The charge information would normally be thought of as a single degree of freedom, so you'd have a 4d nand disk. One reason it is helpful to think of "a dimension" as an independent degree of freedom, is that it becomes a parameter you can focus on improving. So if you say "Well what can we change, or make more precise?", the answer for the nand case is "Well, we can't easily add another spatial degree of freedom, but we can improve our stacking to improve how much we can pack into the vertical dimesion (but we're limited by Job's obsession with thinness). We can improve our measurements of spatial resolution, which will affect 3 of our dimensions, but not the charge. We can also improve our charge resolution. Let's figure out which one is cheapest to scale." Now granted, sometimes tweaking one parameter affects the other, so they aren't strictly independent. You can pack a lot of charge on a NAND, but as the other dimensions get smaller, you start having increased difficulty with leaking. Anyhow, I think there's more here than you're giving them credit for, even if they are guilty of tooting their own horn a bit (shocking). But yes, the article presented here would be greatly improved if it gave examples of how to classify existing technology using the researcher's dimensional classification scheme. It's a common scientific writing tactic that I'm surprised was not used here.
Rygian 5y ago
Which reduces to one single dimension anyway, on a storage medium: my bytes are stored from offset 0 to offset N.
- klodolph 5y ago
  That's a bit reductive.
  - zardo 5y ago
    Maybe for some. For most uses though, people just want to write data to and read it from the storage media. They don't want to think about the coordinates their data was written to on the DVD.
  - seanw444 5y ago
    Ba dum tssss
- 4ad 5y ago
  The fact that spaces of different dimensionality can be put into a bijective map does not mean that dimensionality is not real or useful.
- babypuncher 5y ago
  Yes, the dimensions are an abstraction to make it easier for our human brains to organize certain collections of data. I don't think it makes them any less valid, otherwise we would refer to every data type as an array of bits and nothing more.
arcticbull 5y ago
Isn't this normally referred to as "degrees of freedom" rather than dimensions?
- mvanaltvorst 5y ago
  Statisticians and mathematicians call them dimensions, machine learning engineers call them "features". Statisticians and mathematicians usually interpret observations as vectors in n-dimensional space, which leads to some interesting geometric results. Hence the term dimension, we do mathematics with them as if they were actual points in some high-dimensional space. Degrees of freedom are something a bit more nuanced that are of use when you start to do specific statistical tests, and they do not have to be exactly equal to your amount of dimensions.
- panda-giddiness 5y ago
  No, these are separate concepts. As an example: a simple linear regression through 7 data points has 2 dimensions (x and y) but 5 degrees of freedom (number of data points - number of estimated parameters).
gjm11 5y ago
Sure, "5D" _could_ make sense. It just happens not to in this case. They're mixing up two different sorts of dimensionality that it doesn't make any sense to mix up. 1. How many parameters does it take to describe where you're looking for a given piece of information? This is (kinda) a measure of how many pieces of information there are. Increasing the number of dimensions can mean a very large increase in the amount of data -- imagine going from a 1000x1000 array of things to a 1000x1000x1000 array. The number of dimensions in this case is "two and a bit", because they're using a small number of 2-dimensional layers. 2. How many parameters does it take to describe what you find when you look at a particular piece of information? This is (kinda) a measure of how much information there is in each thing. Increasing the number of dimensions here typically means a rather small increase in the amount of data. E.g., the difference between "1D" and "2D" here would, in the simplest case, be a factor of 2. The number of dimensions in this case is 2, because in each voxel there's a thing parameterized by size and orientation. (I think.) So, three dimensions of the first kind and two of the second. Add 'em up to get 5D, right? Wrong. (At least, if your aim is to inform rather than to deceive.) First problem: these two kinds of dimension do not behave in the same way. Suppose there are N positions along each dimension, and you have A type-1 dimensions and B type-2 dimensions. Then you have N^A places to look for data, and each one stores B log N bits. So adding 1 to A multiplies the amount of data stored by N, typically a very big change; but adding 1 to B multiplies it by (B+1)/B, typically a rather small change. Second problem: the "baseline" is not zero type-2 dimensions but one. You always have at least one type-2 dimension, or else you aren't actually storing any data. So e.g. if you have a grid of memory cells, that's 2 type-1 dimensions and 1 type-2 dimension. Are we going to call this 3D storage? I really hope not. Two-and-a-bit-th problem: as mentioned above, their third type-1 dimension is really hardly there: they have three layers. I guess that is a third dimension, kinda, but it's pretty underwhelming. This is 2-and-a-bit-dimensional storage, with each storage location being able to hold a bit more data because of this size-and-orientation thing. It sounds like it's an impressive technical achievement and might turn out to be a great storage medium. It just isn't in any useful sense five-dimensional. I said at the start that "5D" could make sense. How? Well, suppose you have a system where you have things arranged in a 3D lattice, and then to query each thing you shine light at it and see what it does. And suppose you're able to make each of your thing store different things for different wavelengths of light, and also different things depending on what angle you shine the light at. In that case, the number of type-1 dimensions might be as large as 6: storage locations are indexed by three spatial dimensions, two dimensions describing the direction of the light beam, and one dimension describing the light wavelength. (If you had a system like this, then I bet that in practice those last three dimensions would all be "small", like the third spatial dimension in this silica-glass storage.)

I think Dimension in the data sense of the word, not Dimension in the spatial sense of the word. Multi-dimension data is a thing, and it goes a lot higher than 4, 5, or 7.

gruez 5y ago

>I think Dimension in the data sense of the word Where else is that used? At least to me (and the common layperson), the only usage seems to be from people trying to hype up storage mediums. also, if we're using that definition, does that mean I have a 7D SSD? The NAND cells are arranged on both the planar axis, and stacked (3d NAND). That's 3 dimensions. Each NAND cell also encodes data using various charge levels, which currently tops out at 4 bits. In total that makes for 3 + 4 = 7 dimensions.

kortex 5y ago
That is still O(N^3) in my opinion. If you scale up the media in all directions, your data scales like the cubic. The charge level makes it maybe 3.1 dimensions, but it isn't really scalable. Now if you could add arbitrary voltage levels without losing precision, that would be 4D.
emteycz 5y ago
It's a math/information science term that was applied in physics (its most well known usage), not the other way around...
lelandbatey 5y ago
It's mostly a mathematical term which also applies to pieces of data. See here: https://en.wikipedia.org/wiki/Dimension https://en.wikipedia.org/wiki/Dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
c5e3ebe93d2c 5y ago
In my physics courses we would routinely talk about 6 dimensional spaces when discussing position-momentum space, also we would sometimes refer to two dimensional objects like a moebius strips as "1-dimensional". > also, if we're using that definition, does that mean I have a 7D SSD? Maybe. You can use whatever definitions you want if you feel it's helpful. However, based on your description, if you're using a single bit as a dimension, you should figure out how many bits the the dimensional information encodes. As it stands, it's not clear what you're counting as a 'dimension' in this context. Also, normally we think of degrees of freedom as being independent of each other (hence the claim to 5d in this example). The charge information would normally be thought of as a single degree of freedom, so you'd have a 4d nand disk. One reason it is helpful to think of "a dimension" as an independent degree of freedom, is that it becomes a parameter you can focus on improving. So if you say "Well what can we change, or make more precise?", the answer for the nand case is "Well, we can't easily add another spatial degree of freedom, but we can improve our stacking to improve how much we can pack into the vertical dimesion (but we're limited by Job's obsession with thinness). We can improve our measurements of spatial resolution, which will affect 3 of our dimensions, but not the charge. We can also improve our charge resolution. Let's figure out which one is cheapest to scale." Now granted, sometimes tweaking one parameter affects the other, so they aren't strictly independent. You can pack a lot of charge on a NAND, but as the other dimensions get smaller, you start having increased difficulty with leaking. Anyhow, I think there's more here than you're giving them credit for, even if they are guilty of tooting their own horn a bit (shocking). But yes, the article presented here would be greatly improved if it gave examples of how to classify existing technology using the researcher's dimensional classification scheme. It's a common scientific writing tactic that I'm surprised was not used here.

Rygian 5y ago

Which reduces to one single dimension anyway, on a storage medium: my bytes are stored from offset 0 to offset N.

klodolph 5y ago
That's a bit reductive.
- zardo 5y ago
  Maybe for some. For most uses though, people just want to write data to and read it from the storage media. They don't want to think about the coordinates their data was written to on the DVD.
- seanw444 5y ago
  Ba dum tssss
4ad 5y ago
The fact that spaces of different dimensionality can be put into a bijective map does not mean that dimensionality is not real or useful.
babypuncher 5y ago
Yes, the dimensions are an abstraction to make it easier for our human brains to organize certain collections of data. I don't think it makes them any less valid, otherwise we would refer to every data type as an array of bits and nothing more.

arcticbull 5y ago

Isn't this normally referred to as "degrees of freedom" rather than dimensions?

mvanaltvorst 5y ago
Statisticians and mathematicians call them dimensions, machine learning engineers call them "features". Statisticians and mathematicians usually interpret observations as vectors in n-dimensional space, which leads to some interesting geometric results. Hence the term dimension, we do mathematics with them as if they were actual points in some high-dimensional space. Degrees of freedom are something a bit more nuanced that are of use when you start to do specific statistical tests, and they do not have to be exactly equal to your amount of dimensions.
panda-giddiness 5y ago
No, these are separate concepts. As an example: a simple linear regression through 7 data points has 2 dimensions (x and y) but 5 degrees of freedom (number of data points - number of estimated parameters).

gjm11 5y ago

Sure, "5D" _could_ make sense. It just happens not to in this case. They're mixing up two different sorts of dimensionality that it doesn't make any sense to mix up. 1. How many parameters does it take to describe where you're looking for a given piece of information? This is (kinda) a measure of how many pieces of information there are. Increasing the number of dimensions can mean a very large increase in the amount of data -- imagine going from a 1000x1000 array of things to a 1000x1000x1000 array. The number of dimensions in this case is "two and a bit", because they're using a small number of 2-dimensional layers. 2. How many parameters does it take to describe what you find when you look at a particular piece of information? This is (kinda) a measure of how much information there is in each thing. Increasing the number of dimensions here typically means a rather small increase in the amount of data. E.g., the difference between "1D" and "2D" here would, in the simplest case, be a factor of 2. The number of dimensions in this case is 2, because in each voxel there's a thing parameterized by size and orientation. (I think.) So, three dimensions of the first kind and two of the second. Add 'em up to get 5D, right? Wrong. (At least, if your aim is to inform rather than to deceive.) First problem: these two kinds of dimension do not behave in the same way. Suppose there are N positions along each dimension, and you have A type-1 dimensions and B type-2 dimensions. Then you have N^A places to look for data, and each one stores B log N bits. So adding 1 to A multiplies the amount of data stored by N, typically a very big change; but adding 1 to B multiplies it by (B+1)/B, typically a rather small change. Second problem: the "baseline" is not zero type-2 dimensions but one. You always have at least one type-2 dimension, or else you aren't actually storing any data. So e.g. if you have a grid of memory cells, that's 2 type-1 dimensions and 1 type-2 dimension. Are we going to call this 3D storage? I really hope not. Two-and-a-bit-th problem: as mentioned above, their third type-1 dimension is really hardly there: they have three layers. I guess that is a third dimension, kinda, but it's pretty underwhelming. This is 2-and-a-bit-dimensional storage, with each storage location being able to hold a bit more data because of this size-and-orientation thing. It sounds like it's an impressive technical achievement and might turn out to be a great storage medium. It just isn't in any useful sense five-dimensional. I said at the start that "5D" could make sense. How? Well, suppose you have a system where you have things arranged in a 3D lattice, and then to query each thing you shine light at it and see what it does. And suppose you're able to make each of your thing store different things for different wavelengths of light, and also different things depending on what angle you shine the light at. In that case, the number of type-1 dimensions might be as large as 6: storage locations are indexed by three spatial dimensions, two dimensions describing the direction of the light beam, and one dimension describing the light wavelength. (If you had a system like this, then I bet that in practice those last three dimensions would all be "small", like the third spatial dimension in this silica-glass storage.)