Positive semi definite matrix to LoRa - Understanding AI with Linear Algebra.

This post https://x.com/Mikame_Math/status/2009838510606586249 on Positive Semi definite matrix led me to understand more stuff. I tried to understand if the geometrical shape is related to the epiplexity in computation bound paper, after coming across ellipsoids.

Why math now for an engineer.

how is geometry related to algebra

Geometry and algebra are closely connected because algebra gives geometry a precise language, and geometry gives algebra visual meaning. Here are the key ways they relate:

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1. Coordinate Geometry (Analytic Geometry)

This is the most direct link.

• Geometry studies shapes (lines, circles, curves).

• Algebra describes those shapes using equations.

Examples:

• A line → y = mx + b

• A circle → (x − h)² + (y − k)² = r²

• A parabola → y = ax² + bx + c

By placing shapes on a coordinate plane, geometric problems become algebraic ones.

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2. Solving Geometry Problems with Algebra

Many geometric questions are solved by setting up equations.

Examples:

• Finding the length of a side using the Pythagorean theorem:
  a² + b² = c²

• Finding unknown angles or lengths using systems of equations

• Using formulas (area, volume, distance) that rely on algebra

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3. Visualizing Algebra with Geometry

Geometry helps you see algebra.

Examples:

• A graph shows how an equation behaves

• Intersections of graphs represent solutions to equations

• Slopes represent rates of change

This is why graphing is so important in algebra.

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4. Transformations and Functions

Algebra describes geometric transformations:

• Translation → adding/subtracting values

• Reflection → multiplying by −1

• Rotation → using matrices

• Scaling → multiplying by a constant

Example:

• y = f(x) + 3 moves a graph up

• y = −f(x) reflects it over the x-axis

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5. Linear Algebra and Geometry

Linear algebra generalizes geometry to higher dimensions.

• Vectors represent directions and distances

• Matrices represent transformations (rotation, scaling)

• Planes and lines are described using systems of equations

This is essential in physics, computer graphics, and engineering.

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In short

• Algebra describes geometry

• Geometry visualizes algebra

• Together, they allow us to model and solve real-world problems

If you want, I can explain this with examples at a specific grade level or show how this applies in physics, engineering, or computer graphics.

given this is there an efficient way to do matrix multiplication of AU with algebra faster?

Yes — algebra and geometry together give several efficient ways to multiply matrices like $AU$, depending on what kind of matrix $U$ is and what you need from the result.

Below are the main efficiency ideas, from most practical to more advanced.

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1. Use the geometric meaning of $U$

Matrix multiplication is really applying a transformation.

If

• $A$ is your data (points, vectors, equations), and

• $U$ represents a transformation (rotation, scaling, projection),

then:

Compute the transformation algebraically instead of multiplying blindly.

Examples

• Rotation matrix → use trigonometric identities

• Scaling matrix → just multiply columns or rows

• Projection matrix → often simplifies to dot products

• Orthogonal matrix ($U^TU = I$) → preserves lengths and angles, many terms cancel

This can reduce work from $O(n^3)$ to $O(n^2)$.

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2. Multiply by columns (not entry-by-entry)

Algebraically,

$$AU = [A\mathbf{u}_1 \;\; A\mathbf{u}_2 \;\; \dots \;\; A\mathbf{u}_k]$$

where $\mathbf{u}_i$ are the columns of $U$.

So instead of full matrix multiplication:

• Treat each column of $U$ as a vector

• Multiply $A$ by each vector

This is faster and clearer if:

• $U$ has few columns

• $U$ is sparse (many zeros)

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3. Exploit special structure in $U$

If $U$ has structure, algebra simplifies the multiplication.

Type of $U$	Algebraic shortcut
Diagonal	Scale columns of $A$
Sparse	Ignore zero entries
Triangular	Skip half the operations
Identity	Result is just $A$
Permutation	Just reorder columns

These reduce unnecessary arithmetic.

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4. Associativity trick (when more matrices are involved)

If you have something like:

$$A(UB)$$

Choose the order that is cheapest:

• Multiply smaller matrices first

• Use algebraic dimension analysis to minimize operations

This is called optimal matrix chain multiplication.

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5. Advanced algorithms (theoretical speedups)

For very large matrices:

• Strassen’s algorithm:
  Reduces complexity from $O(n^3)$ to $O(n^{2.81})$

• Coppersmith–Winograd variants:
  Even faster asymptotically (mostly theoretical)

These rely heavily on algebraic identities, not geometry, and are used mainly in high-performance computing.

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6. When geometry really helps

If $U$ represents:

• A basis change

• A projection

• A rotation

Then it’s often better to:

• Transform the vectors first

• Then apply $A$

This avoids full matrix multiplication entirely.

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Summary

Yes — algebra gives faster ways to compute $AU$ by:

• Exploiting structure in $U$

• Interpreting multiplication as transformations

• Avoiding unnecessary operations

• Choosing optimal multiplication order

If you want, tell me:

• The size of $A$ and $U$

• Whether $U$ is diagonal, orthogonal, sparse, etc.

• Whether you need the full product or just $AU\mathbf{x}$

and I can show the most efficient method step by step.

or should I have asked for geometry way

Good question — and the honest answer is: you didn’t ask the wrong thing 🙂

It depends on what you’re trying to do with $AU$.

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Short answer

• If your goal is speed, computation, or implementation → algebra is the right question

• If your goal is understanding, simplification, or insight → geometry is often better

• The best approach uses both

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When algebra is the better way to ask

Ask in algebraic terms if:

• You need to compute $AU$ efficiently

• You care about operation counts

• You’re implementing this in code

• $U$ has exploitable structure (diagonal, sparse, triangular)

Here, geometry helps explain why shortcuts work, but algebra does the work.

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When geometry is the better way to ask

Ask in geometric terms if:

• $U$ represents a transformation (rotation, projection, basis change)

• You want to know what $AU$ does, not just how to compute it

• You’re trying to avoid computing $AU$ at all

Geometric reframing often turns:

$$AU \quad \text{into} \quad \text{“apply } U \text{ to vectors, then apply } A”$$

which can be dramatically simpler.

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Example (why geometry can beat algebra)

Suppose:

• $U$ is an orthogonal matrix (rotation)

• $A$ measures lengths or dot products

Geometric insight:

• Rotations preserve angles and lengths

• So many entries of $AU$ don’t need recomputation

• Sometimes $AU = A$ or differs only by sign or ordering

Algebra alone won’t make that obvious.

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The best question (next time)

Instead of choosing one, ask:

“Can I compute or simplify $AU$ without explicitly multiplying matrices, using algebraic structure or geometric interpretation?”

That invites both tools, which is how mathematicians and engineers actually think.

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Bottom line

• You asked a good algebra question

• A geometric framing might reveal shortcuts

• The fastest solution usually combines both

If you want, give me a concrete example of $A$ and $U$ and I’ll show:

• the algebraic method

• the geometric interpretation

• and which one is actually faster in practice

just checking if we are missing out on an effective way to do the AI matmul

Short answer: no — you’re not missing a hidden “better” method.
For $A I$ (A times the identity), the most effective method is to not do matrix multiplication at all.

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If “AI matmul” means $A \cdot I$

Then algebra and geometry agree perfectly:

• Algebraically:

  $$A I = A$$

  No operations needed.

• Geometrically:
  The identity matrix represents the do-nothing transformation. Applying it changes nothing.

So the optimal algorithm is:

Return $A$ directly.

Anything else is wasted computation.

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If you meant “AI matmul” more generally (A times some special $U$)

Then the situation is:

• There is no universal shortcut better than exploiting:

  • identity

  • sparsity

  • diagonal / orthogonal structure

  • associativity (reordering products)

• Geometry helps you recognize those cases

• Algebra tells you how to exploit them

Once those are exhausted, standard matmul (or optimized BLAS) is already optimal.

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Big-picture takeaway

You’re thinking exactly like a numerical linear algebra person:

• Ask: “Can I avoid doing the multiplication?”

• If yes → geometry / structure

• If no → algebra + optimized kernels

For $A I$, the answer is already as efficient as mathematically possible:
zero multiplications, zero additions.

If you want, tell me what “AI” stands for in your context (identity, attention, adjacency–incidence, etc.), and I’ll double-check that no structural shortcut is being overlooked.

what is positive semi-definiteness of matrices

Positive semidefiniteness (PSD) is a property of certain matrices that guarantees they behave like “non-negative energy” objects. It connects algebra, geometry, and applications very cleanly.

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1. Formal definition (algebraic)

A real $n \times n$ matrix $A$ is positive semidefinite if

$$\mathbf{x}^T A \mathbf{x} \;\ge\; 0 \quad \text{for all vectors } \mathbf{x} \in \mathbb{R}^n.$$

If the inequality is strictly positive for all nonzero $\mathbf{x}$, then $A$ is positive definite.

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2. Required structural properties

For real matrices:

• $A$ must be symmetric

• For complex matrices: $A$ must be Hermitian ($A = A^*$)

If a matrix is not symmetric/Hermitian, it cannot be PSD.

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3. Geometric interpretation

The quantity

$$\mathbf{x}^T A \mathbf{x}$$

represents a quadratic form.

Geometrically:

• PSD matrices define ellipsoids (possibly flattened)

• Directions where $\mathbf{x}^T A \mathbf{x} = 0$ are “flat directions”

• PD matrices give strictly rounded ellipsoids

This is why PSD matrices show up in optimization and physics.

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4. Eigenvalue characterization (most important test)

A symmetric matrix $A$ is PSD if and only if:

$$\text{all eigenvalues of } A \ge 0.$$

• Zero eigenvalues → semidefinite

• All positive eigenvalues → definite

This links PSD directly to geometry: eigenvectors give principal axes.

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5. Factorization viewpoint

A matrix $A$ is PSD iff it can be written as

$$A = B^T B$$

for some matrix $B$.

This explains why:

$$\mathbf{x}^T A \mathbf{x} = \|B\mathbf{x}\|^2 \ge 0$$

This is extremely important in:

• Optimization

• Machine learning

• Statistics

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6. Common equivalent conditions

For a symmetric matrix $A$, the following are equivalent:

• $A$ is PSD

• All eigenvalues ≥ 0

• $A = B^T B$ for some $B$

• All principal minors ≥ 0 (Sylvester’s criterion, semidefinite version)

• $\mathbf{x}^T A \mathbf{x} \ge 0$ for all $\mathbf{x}$

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7. Examples

PSD matrix

$$A = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}$$

Eigenvalues: $0, 3$ → PSD (but not PD)

Not PSD

$$B = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}$$

Has a negative eigenvalue → not PSD

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8. Why PSD matters

PSD matrices appear everywhere:

• Covariance matrices (statistics)

• Gram matrices ($X^TX$)

• Kernel matrices (machine learning)

• Hessians of convex functions

• Energy and stability analysis

• Graph Laplacians

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One-line intuition

A matrix is positive semidefinite if it never turns a vector into “negative energy.”

If you want, I can show:

• how to check PSD efficiently

• why $X^TX$ is always PSD

• how PSD connects to convex optimization

• or how this relates to geometry and transformations

there is a paper on epiplexity that the learning is remembered as geometric shape. Is there an insight there for matmul

Yes — there is a real, non-hand-wavy insight for matrix multiplication, but it’s more about how learning and composition behave, not about a new asymptotically faster matmul algorithm.

I’ll connect the epiplexity / geometric memory idea to matmul in a precise way.

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1. What “learning as geometric shape” usually means (mathematically)

In papers like the one you’re referencing, “learning is remembered as a geometric shape” typically means:

• Learned representations lie on a low-dimensional manifold

• The model’s parameters encode subspaces, cones, or ellipsoids

• Memory is stored as geometry of transformations, not raw numbers

Formally, this often reduces to:

• PSD matrices

• Gram matrices

• Projections

• Compositions of linear maps

All of these are matrix-multiplication-heavy objects.

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2. Key insight for matmul: composition of geometry

Matrix multiplication is composition of linear transformations.

If learning corresponds to a geometric object (subspace, manifold, cone), then:

$$AU \quad \text{is not “numbers × numbers” — it’s “shape after shape.”}$$

That gives three real computational insights.

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3. Insight #1: Low-rank structure (biggest win)

If learned memory lives on a geometric object, it is usually low-rank.

That means:

$$U \approx V W^T \quad \text{with } r \ll n$$

Then:

$$A U = A (V W^T) = (A V) W^T$$

Cost drops from:

$$O(n^3) \;\to\; O(n^2 r)$$

This is huge and already used in:

• LoRA

• SVD compression

• Attention projections

Geometric memory ⇒ low rank ⇒ cheaper matmul

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4. Insight #2: PSD geometry ⇒ factorizations

If learning is stored as a shape, it’s often PSD:

$$U = B^T B$$

Then:

$$A U = A B^T B$$

Instead of one big multiply:

1. Compute $A B^T$

2. Multiply by $B$

This:

• Improves numerical stability

• Allows reuse of $B$

• Enables incremental updates

This is exactly how kernels and covariance memories work.

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5. Insight #3: Learning as subspace rotation, not full transform

If memory is geometric, learning often changes:

• basis

• orientation

• scale along a few directions

Then $U$ is close to:

• Orthogonal

• Block-diagonal

• Sparse in the right basis

So instead of:

$$AU$$

you do:

• Change coordinates

• Apply simple transforms

• Change back

This avoids dense matmul entirely.

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6. What epiplexity adds conceptually

Epiplexity emphasizes:

• Layered geometric complexity

• Emergent structure from composition

Translated to linear algebra:

• Don’t multiply full matrices

• Multiply representations of geometry

• Track how shapes evolve under composition

This doesn’t beat Strassen in theory, but it beats naïve matmul in practice when structure exists.

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7. What it does not give

Important to be honest:

• ❌ No new general-purpose matmul algorithm

• ❌ No asymptotic improvement for arbitrary dense matrices

• ❌ No free lunch if structure is absent

Matrix multiplication lower bounds still apply.

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8. One-sentence takeaway

If learning is stored as geometry, then matrix multiplication should operate on that geometry (rank, subspace, PSD structure) rather than raw entries — and that can make matmul dramatically cheaper when structure exists.

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If you want, you can:

• Point me to the specific epiplexity paper

• Or tell me whether you’re thinking about attention, memory matrices, or weight updates

and I’ll translate the geometric idea into a concrete matmul optimization or reformulation.

is this what the deepseek paper is about

From what’s publicly known so far, the DeepSeek papers are not primarily about a geometric interpretation of learning or a new algebraic matrix-multiplication algorithm—so it’s unlikely they’re directly about “learning remembered as geometric shape” in the sense of a novel algebraic shortcut for matmul.

Here’s a more precise breakdown:

✅ What DeepSeek research actually focuses on

Based on technical reports and analyses:

• Efficient LLM scaling — DeepSeek’s work (e.g., the DeepSeek-V3 technical report) emphasizes architectural and hardware–software co-design to reduce memory and compute costs for huge LLMs. This includes:

  • Multi-head Latent Attention (MLA)

  • Mixture-of-Experts (MoE) sparsity

  • Low-precision training formats like FP8

  • Parallelism strategies optimized for hardware
    These are practical optimizations for training and inference, not fundamentally new geometry-based math.

• Reinforcement learning for reasoning—The DeepSeek-R1 series (e.g., using formally-defined GRPO) focuses on improving reasoning capabilities via RL instead of heavy supervised fine-tuning.

• Sparse and hardware-aligned attention — Some DeepSeek papers (e.g., on Native Sparse Attention) emphasize making attention more efficient and trainable with sparse patterns that match hardware strengths.

• Manifold-Constrained Hyper-Connections (MHC) — Their recent work (via industry discussion) is about constraining internal representational flow to improve stability and training efficiency in deep networks. This is a systems architectural idea, closer to optimization and stability, not a new matmul algebra.

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❓ So is DeepSeek about geometric learning like “epiplexity”?

It doesn’t look like the core DeepSeek research makes that claim explicitly. The core DeepSeek work:

• tackles scaling challenges

• focuses on efficiency, sparsity, and novel training objectives

• explores RL-based reasoning rather than supervised signal scaling
  These are all extremely relevant for AI performance and cost reduction—but they aren’t geometric memory representation theories where the model stores its learning explicitly as shapes/manifolds in a way that gives you a new matmul trick. That’s more typical of research in information geometry, manifold learning, or geometric deep learning, which is a different branch of research.

In other words:

DeepSeek is about how to make LLMs faster and better in practice, not about a new fundamental way to re-interpret matrix multiplication via geometry.

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🧠 Where geometry does intersect with DeepSeek-style work

There are geometric ideas that appear across AI research (and can inspire matmul insights):

• Attention mechanisms can be viewed as computing similarity geometry between tokens.

• Low-rank structure and sparsity embed learned features on lower-dimensional manifolds.

• Mixture-of-Experts implicitly partitions representation space into subspaces.

• Manifold constraints seek stable curves/paths through high-dimensional activation space.

These are geometric in nature, but they don’t replace the fundamental math of linear algebra — they exploit structure within it.

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📌 Bottom line

• No — DeepSeek papers don’t propose a new geometry-based matmul algorithm as far as the public technical reports show.

• They do focus on efficient computation, sparsity, and architectural optimization.

• Geometry does show up conceptually, but in the context of representation and stability, not as a replacement for standard algebraic matmul.

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If you want, I can help connect specific ideas from DeepSeek’s architecture or training innovations to your earlier thoughts about geometry and matmul (e.g., what part of MLA or Mixture-of-Experts is geometric or low-rank) — just let me know which part interests you most!

you said mainfold and hence I thought they might be related.

That’s a very reasonable connection to make — and you weren’t “off” at all. The key is what kind of manifold is being talked about, and at what level.

Let me disentangle it cleanly.

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1. Why “manifold” made it sound related

When you hear:

• learning is geometric

• representations lie on manifolds

• manifold-constrained training

it’s natural to think:

“Ah — maybe this changes matrix multiplication itself.”

That would be a big deal. But in DeepSeek (and most modern LLM work), the manifold lives in representation space, not in the algebra of matmul.

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2. Two very different meanings of “manifold”

(A) Manifold of representations (DeepSeek, LLMs)

This is what DeepSeek is doing.

• Activations of a network lie on a low-dimensional manifold inside a huge vector space

• Training tries to:

  • keep representations stable

  • guide them along smooth trajectories

  • avoid chaotic drift

Matrix multiplication is still:

$$Y = XW$$

Nothing about how matmul is computed changes.

👉 Geometry here is about where the vectors live, not how we multiply matrices.

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(B) Manifold of operators (what you were intuitively asking)

This is closer to epiplexity and your matmul question.

Here the idea would be:

• Learned matrices themselves live on:

  • low-rank manifolds

  • PSD cones

  • Grassmannians (spaces of subspaces)

• Learning is moving along a geometric space of operators

This can affect how we do matmul, because:

• You multiply factorized objects

• You compose structured transforms

• You avoid dense products

DeepSeek is not primarily doing this.

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3. Why DeepSeek uses “manifold” anyway

In DeepSeek-style work, “manifold” is mostly shorthand for:

• controlling optimization geometry

• improving training stability

• enabling sparsity and expert routing

• keeping activations well-behaved

It’s closer to dynamical systems and optimization geometry than to linear-algebraic reformulation.

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4. Where the overlap actually is

There is a narrow overlap — just not a revolutionary one.

• Low-rank structure → fewer effective matmuls

• Sparse routing → fewer matmuls executed

• Shared subspaces → reuse projections

These reduce how many matmuls you do, not how matmul itself works.

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5. Why your intuition is still good

Your line of thought is exactly how real advances happen:

• “If learning is geometric…”

• “…why are we multiplying dense matrices blindly?”

That question has already led to:

• LoRA

• MoE

• Low-rank attention

• Kernel methods

• Factorized parameterizations

DeepSeek is downstream of those ideas, not the source of a new algebra.

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6. One-sentence resolution

DeepSeek talks about manifolds of activations, whereas your question is about manifolds of linear operators — only the latter gives direct leverage on matrix multiplication.

If you want, next we can:

• look at operator manifolds (Grassmannians, PSD cones)

• map epiplexity ideas to factorized matmul

• or identify the exact point where geometry could still give a matmul breakthrough

You’re asking the right kind of question — just at a deeper layer than DeepSeek is working on.

this is the paper - From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence

Yes — I found the actual paper you’re talking about. It’s From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence, and its core idea helps clarify where geometry, information structure, and learning intersect — but not in the way that directly gives a new matrix-multiplication algorithm.

Here’s what the paper actually says and how it relates to your earlier questions:

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📌 What the Epiplexity Paper Is About

Core idea

The paper proposes a new measure, epiplexity, which captures the structural, learnable information in data for a computationally bounded learner — i.e., a model with limited time/compute — unlike classical measures like Shannon entropy or Kolmogorov complexity that assume unlimited computation.

• It explains how deterministic computation can “create” useful information for bounded learners even though classical information theory says information cannot increase under deterministic transformations.

• It introduces time-bounded entropy (the unpredictable parts for a learner with limited compute) and epiplexity (the structural parts the learner can extract).

So instead of asking “how much data contains information” in the traditional sense, it asks:

“How much useful, learnable structure can a real learner extract with limited compute?”

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📌 How this relates to “geometric” views of learning

The paper is not about geometric manifolds in the linear algebra sense (like subspaces or operator geometry). Instead, it’s about information geometry in the sense of structure and predictability:

• “Structure” here is about patterns that a model can compress and generalize, not about geometric shapes in vector space or manifolds of operators.

• It uses coding theory (minimum description length) and computational constraints to define what is learnable, rather than what is merely present in the raw data.

This is conceptually related to manifold ideas — because both are about “structure beyond randomness” — but they are operating in different spaces:

• In geometry/manifold learning (like UMAP or embeddings), we think of data points lying on a geometric surface in high-dimensional space.

• In epiplexity, we think about learnable structure vs. unpredictable noise relative to compute bounds.

So both are concerned with structure, but at different levels:

Concept	Focus	Relation to Matmul
Manifolds in geometry	Data/representations lie on smooth surfaces	Can imply low-dimensional subspaces → matmul shortcuts like low-rank
Epiplexity	What patterns a bounded learner can extract	Structures learning and generalization, not matmul cost
Algebraic matmul efficiency	Reducing arithmetic for matrix multiplies	Practical computation cost

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📌 What epiplexity does not propose

It does not propose:

✅ A new computational model of matrix multiplication
❌ A geometry-based replacement for standard algebraic matmul
⚠️ A direct speedup over known algorithms like BLAS / Strassen / block-mul

Instead, it provides a theoretically grounded way to think about:

• What information matters for learning

• How data order and deterministic computation affect what can be learned

• Why some datasets generalize better (i.e., have higher epiplexity)

These are deep insights into learning theory, not surprising computational shortcuts for basic algebra like matmul.

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📌 Where the connection is

The paper does indirectly touch on deeper structure in representations:

• Recognizing that models internalize structure beyond raw statistical patterns

• Understanding that learnable patterns are effectively representations on some manifold of internal structure

• Establishing that structural complexity (epiplexity) matters more than mere entropy for generalization

That connects with your earlier intuition — that learning and geometry are related — but it does not reduce to matrix multiply shortcuts. Instead, the connection is:

High-epiplexity data → richer internal representations → potentially structured matrices (low-rank, manifold-aligned) → practical computational benefits only when you exploit that structure.

So the epiplexity idea motivates why you could want geometry-aware computation, but it does not itself give an algorithm that makes matmul faster.

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🧠 In other words

• Epiplexity says: what matters for learning is not raw data entropy, but learnable structure under compute limits.

• This motivates geometric/structured representations in models.

• Having geometric structure can lead to efficient computation (e.g., low-rank matmul), but epiplexity by itself is not the algorithmic tool.

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📎 If you want to go further

I can help you with:

1. How epiplexity might predict which matrix structures arise in learned representations

2. How to exploit structure in learned weight matrices for efficient matmul

3. Examples where epiplexity correlates with practical model performance

Just tell me what direction you want next!

why is matrice PD important

Positive-definite (PD) matrices are important because they are the mathematical objects that guarantee stability, uniqueness, and well-behaved geometry. They show up whenever you want something to be strictly meaningful: distances, energy, curvature, certainty, or convexity.

I’ll explain this from several angles and then tie it back to learning and geometry (including epiplexity-style thinking).

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1. Core definition (one line)

A symmetric matrix $A$ is positive definite if

$$\mathbf{x}^T A \mathbf{x} > 0 \quad \text{for all } \mathbf{x} \neq 0$$

This means no direction is flat or negative.

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2. Geometry: PD = real distance and shape

A PD matrix defines a true geometry.

• $\mathbf{x}^T A \mathbf{x}$ is a squared distance

• The set $\mathbf{x}^T A \mathbf{x} = 1$ is an ellipsoid

• No direction collapses to zero

If a matrix is only PSD:

• Some directions are flat → information is lost

If it’s indefinite:

• Some directions flip sign → geometry breaks

This is why:

• PD ⇒ valid metric

• PSD ⇒ degenerate metric

• Indefinite ⇒ not a metric

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3. Optimization: PD = unique, stable solutions

In optimization:

• Hessian PD ⇒ strictly convex function

• Strict convexity ⇒ unique minimum

• Gradient descent converges reliably

Example:

$$f(x) = \tfrac{1}{2} x^T A x$$

• If $A$ is PD → one stable minimum

• If $A$ is PSD → flat directions

• If $A$ is indefinite → saddle points

This is huge in ML training and control systems.

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4. Learning & statistics: PD = meaningful uncertainty

In statistics and ML:

• Covariance matrices must be PSD

• Precision matrices (inverse covariance) must be PD

Why?

• PD ⇒ every variable has positive variance

• Invertible ⇒ you can condition, update, and infer

Gaussian models require PD covariance.

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5. Linear algebra: PD = invertibility + stability

PD matrices give you:

• Guaranteed invertibility

• Stable numerical behavior

• Well-conditioned systems (if eigenvalues aren’t tiny)

Solving:

$$Ax = b$$

is far safer if $A$ is PD.

This is why:

• Cholesky decomposition works only for PD matrices

• Many solvers assume PD by default

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6. Energy & physics: PD = physical stability

In physics and engineering:

• PD matrices represent energy

• Negative directions ⇒ instability

• Zero directions ⇒ unconstrained motion

Mechanical systems, electrical networks, and control laws rely on PD matrices to ensure systems don’t blow up.

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7. Why PD matters more than PSD in practice

PSD is often not enough:

Property	PSD	PD
Non-negative energy	✅	✅
No flat directions	❌	✅
Invertible	❌	✅
Unique optimum	❌	✅
Stable geometry	⚠️	✅

In learning:

• PSD = representation exists

• PD = representation is fully informative

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8. Connection to geometry & epiplexity

Here’s the deeper link to your earlier questions:

• Epiplexity is about extractable structure

• Structure lives in geometry

• Geometry needs PD metrics to be meaningful

If a learned representation collapses (PSD but not PD):

• The model can’t distinguish directions

• Information is present but not extractable

So:

PD matrices guarantee that learned structure is usable, not just present.

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9. One-sentence intuition

A positive-definite matrix ensures that every direction matters, every distance is real, and every solution is stable.

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If you want, I can next explain:

• PD vs PSD in neural network Hessians

• Why attention kernels must be PSD

• How PD constraints improve training

• Or how PD geometry relates to memory and representation

Just tell me where you want to go next.

Sure all of these.

Perfect! Let’s go step by step — we’ll connect PD vs PSD, neural network Hessians, attention kernels, training stability, and memory/representation geometry. I’ll also keep it intuitive, then show how it ties back to your epiplexity/geometric intuition.

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1. PD vs PSD in Neural Network Hessians

Hessian basics

• The Hessian $H$ of a neural network loss $L(\theta)$ is a matrix of second derivatives:

$$H = \nabla_\theta^2 L(\theta)$$

• Its entries tell us the curvature of the loss landscape.

PD vs PSD

Hessian Type	Interpretation
PD	All directions are “uphill” from a minimum ⇒ strictly convex locally ⇒ unique local minimum in that neighborhood.
PSD	Flat directions exist ⇒ some parameters do not affect loss ⇒ degenerate minima or plateaus.
Indefinite	Some directions downhill, some uphill ⇒ saddle points.

Why it matters in NN training:

• If $H$ is PD: Newton’s method converges quickly.

• If $H$ is PSD but not PD: optimizer might wander in flat directions.

• If $H$ is indefinite: optimizer can get stuck at saddles or oscillate.

✅ Key insight: For well-behaved learning, we want local PD curvature around minima — it ensures stable, usable gradients.

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2. Attention kernels must be PSD

In transformers, attention is often written as:

$$\text{Attention}(Q,K) = \text{softmax}\left(\frac{Q K^T}{\sqrt{d}}\right)$$

• $K K^T$ or $Q K^T$ often forms a Gram matrix.

• Gram matrices are always PSD because:

$$\mathbf{x}^T (K K^T) \mathbf{x} = \|K^T \mathbf{x}\|^2 \ge 0$$

Why this matters:

1. PSD ensures that “similarity scores” are non-negative in the quadratic form sense.

2. PD (if achieved) would guarantee strict separation of directions, avoiding degenerate attention where two keys collapse into one.

3. Practically, PD or near-PD helps stability in gradient propagation, particularly in deep transformers.

✅ Key insight: PD/PSD in attention = well-conditioned similarity geometry, avoiding degenerate memory representations.

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3. PD and training stability

Several mechanisms in training rely on PD matrices:

1. Covariance-based optimizers (e.g., Adam, RMSProp)

   • Compute moving averages of squared gradients → matrices that should be PSD.

   • PD ensures proper preconditioning without dividing by zero.

2. Natural gradient methods

   • Use the Fisher Information Matrix (FIM) → always PSD, ideally PD.

   • PD guarantees invertibility, so you can compute $\theta \leftarrow \theta - \eta F^{-1} g$ safely.

3. Second-order updates

   • Hessians must be PD for Newton updates to converge; if not, updates can diverge along negative curvature directions.

✅ Key insight: PD = optimizer can safely trust curvature information. PSD alone may give flat directions that stall learning.

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4. PD geometry and memory / representation

When thinking of representations as geometric objects (your earlier epiplexity/manifold intuition):

• PSD matrix → some directions may carry zero variance → information loss.

• PD matrix → every direction has meaningful variance → full, extractable information.

Example in memory systems:

• Let $M$ be a learned memory covariance or kernel matrix.

• If $M$ is PSD but not PD:

  • Some memory directions collapse → model “forgets” certain patterns.

• If $M$ is PD:

  • Every direction can store distinct information → maximum representational capacity.

Connection to epiplexity:

• Epiplexity measures extractable learnable information.

• PD matrices in representation space ensure that the model can actually extract and use all the information — none is “degenerate” or inaccessible.

✅ Key insight: PD matrices = usable memory / usable geometry.

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5. Summary Table: PD Across ML Concepts

Concept	PSD	PD	Why PD matters
Hessian	Flat directions	Strict curvature	Stable, unique minima
Attention / Gram	Non-negative similarity	All directions distinct	Avoid degenerate attention, better gradients
Optimizers	Preconditioning okay	Invertible, stable	Safe second-order updates
Representations / Memory	Some directions unused	Full representational capacity	Extractable memory, no collapse
Epiplexity	Learnable structure may be partial	Fully usable structure	Ensures all learnable info is preserved

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6. One-line intuition tying it together

Positive-definite matrices guarantee every direction is meaningful, stable, and usable, which underlies stable learning, faithful attention, reliable optimization, and full representational capacity — essentially ensuring that nothing in the model’s learned “geometry” is wasted or degenerate.

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If you want, I can draw a visual diagram showing:

• PSD vs PD Hessian geometry

• Attention kernel directions

• Memory representation ellipsoids

…all in one image, so you can see why PD matters visually in ML.

Do you want me to make that diagram?

I am also curious about the Lora and other stuff you mentioned as a result of passing this way.

Perfect! Let’s dive into LoRA, low-rank representations, and related methods — and I’ll show you how they connect to matrix geometry, PD/PSD ideas, and efficiency in neural networks. I’ll break it down in layers so it all ties together.

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1. What LoRA is

LoRA = Low-Rank Adaptation. It’s a method to fine-tune large pre-trained models efficiently:

• Instead of updating the full weight matrix $W \in \mathbb{R}^{d \times d}$ during fine-tuning, you learn low-rank matrices $A, B$ such that:

$$\Delta W = A B$$

where $A \in \mathbb{R}^{d \times r}, B \in \mathbb{R}^{r \times d}$ and $r \ll d$.

• The new effective weight is:

$$W' = W + \Delta W = W + A B$$

Key point: You only learn a small number of parameters ($2 \cdot d \cdot r$ instead of $d^2$) → huge efficiency gain.

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2. Why LoRA works: the geometric intuition

Think back to our PD/PSD and manifold discussion:

• Weights as linear maps are like transforming geometric space.

• LoRA assumes that the effective change in the transformation lies in a low-dimensional subspace.

• Instead of updating all directions, you only need to tweak a few “principal directions” where learning is most effective.

Geometrically:

• Original $W$ defines some shape / subspace of representations.

• LoRA updates tweak just the important directions, leaving the rest untouched.

• This is why learning can still be effective with far fewer parameters.

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3. Connection to PD/PSD and low-rank structure

• Many learned matrices are PSD or approximately PSD (like attention kernels or Gram matrices).

• If $W$ or $\Delta W$ is low-rank:

$$\Delta W = A B$$

• You can think of it as moving along a submanifold of the full weight space — only the directions that matter for generalization.

• In Hessian terms: LoRA changes mainly directions corresponding to large eigenvalues (high curvature) — more efficient than touching flat directions (small eigenvalues).

✅ Insight: Low-rank updates + PD/PSD structure = efficient, stable learning along the “meaningful geometry”.

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4. Other related methods

1. Low-Rank Factorization

   • Factorize large matrices: $W \approx U V^T$

   • Reduces memory and compute

   • Exploits low-dimensional structure of learned transformations

2. Mixture-of-Experts (MoE)

   • Each layer has multiple “expert” weight matrices

   • Only a subset is active per input → sparse matrix multiply

   • Efficiently leverages manifold/clustered structure in representation space

3. LoRA + MoE

   • LoRA can be applied per expert → fine-tune efficiently within sparse routing

   • Still modifies only the important subspace of each expert

4. Structured Sparse Updates

   • Update only blocks of matrices along high-curvature directions

   • Geometric interpretation: learning along tangent planes of the weight manifold

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5. Why this connects to Epiplexity / geometric ideas

• Epiplexity measures learnable structure in data under bounded computation.

• LoRA, low-rank factorization, and MoE assume the same principle for models themselves:

  Only a small, structured subspace of the full parameter space matters for the task.

• From a geometry perspective:

  • Full matrix = full $d^2$-dimensional space

  • Low-rank = a manifold of dimension $2 r d$ along which learning occurs

  • PD/PSD structure ensures updates are meaningful, not degenerate

✅ Intuition: LoRA is practically exploiting the same “epiplexity of the model” — only the learnable, non-flat directions are adjusted.

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6. Visual / intuitive analogy

Think of the weight matrix as a huge landscape:

• Full matrix update = walk everywhere → expensive

• Low-rank update = walk along the main valley directions → efficient

• PSD/PD ensures valleys are well-defined, not flat or upside-down

• Epiplexity = tells you which valleys contain learnable information

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7. Computational benefits

Method	Parameters updated	Compute reduction	Geometric intuition
Full fine-tune	$d^2$	none	moves entire space
LoRA	$2 d r$	massive	moves only principal directions
MoE	subset of experts	sparse	only active subspaces
Low-rank factorization	$d r + r d$	linear instead of quadratic	only “meaningful manifold”

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✅ Key takeaway:

LoRA and these methods are practical ways to exploit geometric/low-rank structure in neural networks, which directly connects to:

• PD/PSD ideas (stable, non-degenerate directions)

• Manifold/epiplexity intuition (only some subspaces carry meaningful, learnable information)

• Efficient matrix multiplication (multiply only along the low-rank subspace)

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If you want, I can draw a single diagram showing:

• A full weight matrix

• Its low-rank LoRA update

• How it corresponds to a manifold of meaningful directions

• Connection to PD/PSD curvature and memory geometry

It will make all these abstract concepts immediately visual and intuitive.