Block Number Formats are (Still!) Direction Preservers

In my previous post, I argued that block number formats can be understood geometrically as direction preservers. That argument relied on an idealization: once a block direction had been chosen, its scale could be set optimally as an arbitrary real number.

Real hardware formats do not usually work that way. In many practical schemes, block scales are quantized very coarsely, sometimes all the way down to powers of two. In particular, in the MX specification, all the concrete compliant formats use E8M0 scaling.

So does the directional picture I painted in my last post survive this brutal scaling? Here I will argue, in the first of what I hope will be a short sequence of follow-up blog posts, that it does.

From ideal block scales to quantized block scales

Recall the setup from the earlier post. A vector is partitioned into blocks, $v = (v_1,\dots,v_B)$ , and each block is approximated as $\hat v_b = \beta_b m_b$ , where $m_b$ is a low-precision mantissa vector and $\beta_b$ is a scalar block scale.

In the earlier post, I assumed that $\beta_b$ was an arbitrary real value, chosen optimally in the least-squares sense. That gave the ideal blockwise representation $\hat v$ .

Now let us keep the same mantissa vectors $m_b$ , but suppose that the scale factors themselves must be quantized. Write the implemented scale as $\tilde \beta_b$ , so that the represented block becomes $\tilde v_b = \tilde \beta_b m_b$ .

It is convenient to define the multiplicative scale error $x_b = \frac{\tilde \beta_b}{\beta_b}$ . Then $\tilde v_b = x_b \hat v_b$ .

Note that, of course, quantizing the block scale does not change the chosen direction of a block at all; it only changes its length. So the only directional distortion comes from the relative rescaling of different blocks.

An exact cosine formula

Let $\alpha_b = \frac{\|\hat v_b\|^2}{\sum_j \|\hat v_j\|^2}$ , so that $\alpha_b$ is the fraction of the ideal projected vector’s energy contained in block $b$ .

Then it can be shown that $\cos(\hat v,\tilde v) = \frac{\sum_b \alpha_b x_b}{\sqrt{\sum_b \alpha_b x_b^2}}$ (the proof is included at the end of this post).

So the effect of scale quantization on direction depends only on how uneven the factors $x_b$ are across blocks. If all blocks were rescaled by the same factor, direction would be unchanged.

Exponent-only power-of-two scaling

Now consider the coarsest plausible case: each block scale is rounded to the nearest power of two. Then each multiplicative error satisfies $x_b \in [2^{-1/2},\,2^{1/2}]$ .

So, from our exact cosine formula, we are interested in how small $\frac{\sum_b \alpha_b x_b}{\sqrt{\sum_b \alpha_b x_b^2}}$ can be, when all the $x_b$ lie in the interval $[2^{-1/2},\,2^{1/2}]$ .

A simple inequality shows that the answer depends only on the two extreme values of the interval. If all the block rescaling factors lie in $[\ell,u]$ , then

$\frac{\sum_b \alpha_b x_b}{\sqrt{\sum_b \alpha_b x_b^2}}\ge \frac{2\sqrt{\ell u}}{\ell+u}$ (proof at the end of this blog post).

In the power-of-two case we have $\ell=2^{-1/2}$ , $u=2^{1/2}$ , so $\ell u=1$ , and therefore

$\cos(\hat v,\tilde v)\ge \frac{2}{2^{-1/2}+2^{1/2}} = \frac{2\sqrt2}{3}\approx 0.943$ .

Equivalently, $\angle(\hat v,\tilde v)\le 20^\circ$ .

So even if every block scale is rounded to the nearest power of two, the resulting vector remains within about $20^\circ$ of the ideally scaled one.

That is the main result of this post.

One striking feature of the bound is that it does not depend on the dimension of the vector. The reason is that the worst case is already attained by a two-group energy split: some blocks rounded up, others rounded down. Once those two groups exist, adding more blocks or more dimensions does not make the bound worse, as is apparent from the proof below.

20 degrees is less than it sounds

Our everyday intuition may tell us that this angle is not huge, but it’s not that small either. In a sense, that’s true. But angles behave very differently in high-dimensional spaces. In high dimension, most random vectors are almost orthogonal to one another: their angle is close to $90^\circ$ , so a guarantee that an approximation remains within $20^\circ$ of the original vector is much stronger than it would sound in two or three dimensions.

Beyond power-of-two

We’ve analysed power-of-two scaling here for two reasons: because it’s in a sense the crudest possible floating-point rounding, and because it’s commonly used in real hardware designs.

That does not mean it’s optimal. But it does raise two further questions. Firstly, we’ve assumed here that the exponent range is sufficiently wide – what if it’s not? Secondly – and relatedly – how much better can this angular bound get by spending some of the scale bits on greater precision?

My view is that the answer becomes clearer once a tensor-wide high-precision scale is introduced, something NVIDIA has recently done. In that setting, the block scales get relieved of their additional duty to capture global magnitude. This will be the subject of the next post on the topic!

Proofs

Readers not interested in the algebra can safely skip this section.

Cosine formula

Recall that $\tilde v_b = x_b \hat v_b$ for each block $b$ .

Then, because the blocks occupy disjoint coordinates, $\langle \hat v,\tilde v\rangle = \sum_b \langle \hat v_b,\tilde v_b\rangle = \sum_b x_b \|\hat v_b\|^2$ .

Therefore $\cos(\hat v,\tilde v) = \frac{\langle \hat v,\tilde v\rangle}{\|\hat v\|\,\|\tilde v\|} = \frac{\sum_b x_b \|\hat v_b\|^2}{\sqrt{\sum_b \|\hat v_b\|^2}\sqrt{\sum_b x_b^2 \|\hat v_b\|^2}}$ .

Now, as per the main blog post, define $\alpha_b = \frac{\|\hat v_b\|^2}{\sum_j \|\hat v_j\|^2}$ .

Writing $S=\sum_j \|\hat v_j\|^2$ , so that $\|\hat v_b\|^2=\alpha_b S$ , the numerator becomes $S\sum_b \alpha_b x_b$ and the denominator becomes $S\sqrt{\sum_b \alpha_b x_b^2}$ , giving

$\cos(\hat v,\tilde v)=\frac{\sum_b \alpha_b x_b}{\sqrt{\sum_b \alpha_b x_b^2}}$ .

$\square$

20 degree bound

Assume that all the multiplicative error factors lie in an interval $x_b \in [\ell,u]$ with $u > \ell > 0$ .

Let $\mu := \sum_b \alpha_b x_b,\qquad q := \sum_b \alpha_b x_b^2$ .

Then the cosine is just $\mu/\sqrt q$ . Since each $x_b\in[\ell,u]$ , we have

$(x_b-\ell)(x_b-u)\le 0$ .

Expanding this gives

$x_b^2 \le (\ell+u)x_b - \ell u$ .

Multiplying by $\alpha_b$ and summing over $b$ gives

$q \le (\ell+u)\mu - \ell u$ .

Therefore $\frac{\mu^2}{q}\ge \frac{\mu^2}{(\ell+u)\mu-\ell u}$ .

Now the weighted mean $\mu$ also lies in the interval $[\ell,u]$ , so it remains to minimize

$\frac{\mu^2}{(\ell+u)\mu-\ell u}$ over $\mu\in[\ell,u]$ .

Differentiating shows that the minimum occurs at $\mu=\frac{2\ell u}{\ell+u}$ , the harmonic mean of $\ell$ and $u$ .

Substituting this value gives

$\frac{\mu^2}{q}\ge \frac{4\ell u}{(\ell +u)^2}$ ,

and therefore

$\frac{\mu}{\sqrt q}\ge \frac{2\sqrt{\ell u}}{\ell +u}$ .

So we have proved that

$\frac{\sum_b \alpha_b x_b}{\sqrt{\sum_b \alpha_b x_b^2}}\ge \frac{2\sqrt{\ell u}}{\ell+u}$ .

Finally, in the power-of-two case we have

$\ell=2^{-1/2}$ and $u=2^{1/2}$ , so $\ell u=1$ , and hence

$\cos(\hat v,\tilde v)\ge \frac{2}{2^{-1/2}+2^{1/2}} = \frac{2\sqrt2}{3}$ .

Numerically,

$\frac{2\sqrt2}{3}\approx 0.943$ ,

$\angle(\hat v,\tilde v)\le \arccos\left(\frac{2\sqrt2}{3}\right) < 20^\circ$ .

$\square$

Block Number Formats are Direction Preservers

I’ve recently returned from the SIAM PP 2026 conference and as always, conferences help provide time for research reflection. One thing I’ve been reflecting on during my journey back is the various explanations people give for why the machine learning world is so keen on block number formats (MX, NVFP, etc.) – see my earlier blog post on MX if you need a primer. Many hardware engineers tend to answer that they lead to efficient storage, or efficient arithmetic, or improved data transfer bandwidth, which are all true. But I think there’s another complementary answer that’s less well discussed (if indeed it is discussed at all). I hope this blog post might help stimulate some discussion of this complementary take.

On the numerical side, at first glance it might seem surprising that despite these formats representing numbers with very limited precision, large neural networks often tolerate them remarkably well, with little loss in accuracy. In my experience, most explanations focus on dynamic range, quantization noise, the inherent noise robustness of neural networks, or calibration techniques. But I suspect there is also a simple geometric way to think about what these formats are doing: Block number formats help preserve vector direction. And for many machine learning computations, preserving direction matters far more than preserving exact numerical values.

Block formats inherently represent direction and magnitude

Consider a vector $v$ whose coordinates are partitioned into blocks $v = (v_1, v_2, \dots, v_B)$ .

In a block format, each block is represented using a shared scale and low-precision mantissas. For ease of discussion, we’ll consider the simplest case here, where scales are allowed to be arbitrary real-valued. In general, they may be much more restricted, e.g. powers of two.

Each block is approximated as $\hat v_b = \beta_b m_b$

where

$m_b$ is a vector of low-precision mantissas, and

$\beta_b$ is a scalar shared scaling factor.

In other words, each block can be thought of as a direction (encoded by the mantissas) multiplied by a magnitude (the shared scale). Strictly speaking, the mantissa vectors $m_b$ need not be normalized, and in many formats their entries may have quite different magnitudes (for example in integer mantissa formats such as MXINT). However this does not change the geometry. The representation $\hat{v}_b = \beta_b m_b$ is invariant to rescaling of $m_b$ : multiplying $m_b$ by any constant simply rescales $\beta_b$ by the inverse factor. What matters for the approximation is therefore only the direction of $m_b$ , i.e. the one-dimensional subspace it spans.

Often we don’t think of it like this, but broadly speaking this is what has happened: block scaling allows us to decouple magnitude and direction representation. This resembles the familiar decomposition $v = \|v\|\frac{v}{\|v\|}$ of a vector into its magnitude and direction, but applied locally within blocks.

If the mantissa vector $m_b$ points roughly in the same direction as the original block $v_b$ , then scaling it appropriately produces a good approximation of that block.

OK, but does preserving directions block by block actually preserve the direction of the whole vector? It turns out that the answer is yes.

Direction Preservation

Let us make the reasonable assumption that the scale of each block $\beta_b$ is not chosen arbitrarily, but rather is the best possible scale for that block in the least squares sense, for whatever mantissa vector we choose, i.e. $\beta_b = \arg\min_{\beta} \|v_b - \beta m_b\|^2$ . Then $\hat v_b$ is the orthogonal projection of $v_b$ onto the line spanned by $m_b$ .

So to what extent do the approximate and the original block vector point in the same direction? We can measure the block cosine similarities of the blocks as: $\rho_b = \frac{\langle v_b,\hat v_b\rangle}{\|v_b\|\|\hat v_b\|}$ .

Equally, we can measure the the cosine similarity of the full vectors (the concatenation of the original blocks versus the concatenation of the approximated blocks): $\rho = \frac{\langle v,\hat v\rangle}{\|v\|\|\hat v\|}$ .

My aim here is to explain why small error in direction at block level leads to small error at vector level.

First, let’s define $w_b = \frac{\|v_b\|^2}{\|v\|^2}$ , which we can think of as the fraction of the vector’s energy contained in block $b$ ; these add to 1 over the whole vector. Now we can state the result:

Theorem (Block Cosines)

Under the blockwise least-squares scaling, $\rho = \sqrt{\sum_{b=1}^{B} w_b \rho_b^2 }$ .

For proof, see end of post.

In simple terms, this theorem states that the cosine similarity of the whole vector is the energy-weighted RMS of the block cosine similarities.

What are the implications?

The weights $w_b$ represent how much of the vector’s energy lies in each block. Blocks that contain very little energy contribute very little to the final direction. The important consequence is that direction errors do not accumulate catastrophically across blocks. Instead, the overall directional error simply depends on a weighted average of the block direction errors. In other words, if block number formats preserve the directions of individual blocks, they automatically preserve the direction of the entire vector.

Many core operations in machine learning depend heavily on vector direction. Notably, during training, stochastic gradient descent updates are already in the form of magnitude (learning rate) + direction. We already have a knob controlling magnitude (the learning rate); what matters is that the direction is preserved. In attention mechanisms and embedding, directional similarity measures are very important. Even for the humble dot product, the workhorse of inference, preservation of direction means that small perturbations in input give rise to only small perturbations in output, so the dot product behaves robustly.

Conclusion

Block floating-point and similar formats like block mini-float, MX, NVFP, are usually explained in terms of dynamic range and quantization noise. But geometrically, I like the perspective that they do something simpler: they approximate each block of a vector as direction × magnitude.

And as long as the block directions are preserved reasonably well, the direction of the whole vector is preserved too.

I think this is a useful intuition as to why very low-precision formats can work so well in modern machine learning systems. Block number formats are, in a very real sense, direction preservers. From this perspective, such low-precision block formats succeed not because they represent individual numbers accurately, but because they preserve the geometry of vectors.

Lots of extensions of this kind of analysis are of course possible. To name just a few:

We’ve focused on vectors, but tensor-level scaling may have interesting interplay with batching during training, for example
We made the simplifying assumption that scaling factors were real valued, but these can be restricted, most significantly to powers of two, and the analysis would need to be modified to incorporate that change.
We’ve not discussed mantissas at all, lots more of interest could be said here.
Potentially this approach could help provide some guidance to the empirical sizing of blocks in a block representation.

If anyone would like to work with me on this topic, do let me know your ideas.

Proof of the theorem

Readers not interested in the algebra can safely skip this section.

For each block $b$ , the approximation $\hat v_b = \beta_b m_b$ with $\beta_b$ chosen by least squares is the orthogonal projection of $v_b$ onto the line spanned by $m_b$ .

So we can write $v_b = \hat v_b + r_b$ where $r_b$ is orthogonal to $\hat v_b$ .