Learning on LoRAs: GL-Equivariant Processing of Low-Rank Weight Spaces for Large Finetuned Models

## Abstract

Abstract Low-rank adaptations (LoRAs) have revolutionized the finetuning of large foundation models, enabling efficient adaptation even with limited computational resources. The resulting proliferation of LoRAs presents exciting opportunities for applying machine learning techniques that take these low-rank weights themselves as inputs.
In this paper, we investigate the potential of Learning on LoRAs (LoL), a paradigm where LoRA weights serve as input to machine learning models.
For instance, an LoL model that takes in LoRA weights as inputs could predict the performance of the finetuned model on downstream tasks, detect potentially harmful finetunes, or even generate novel model edits without traditional training methods. We first identify the inherent parameter symmetries of low rank decompositions of weights, which differ significantly from the parameter symmetries of standard neural networks.
To efficiently process LoRA weights, we develop several symmetry-aware invariant or equivariant LoL models, using tools such as canonicalization, invariant featurization, and equivariant layers.
We finetune thousands of text-to-image diffusion models and language models to collect datasets of LoRAs. In numerical experiments on these datasets, we show that our LoL architectures are capable of processing low rank weight decompositions to predict CLIP score, finetuning data attributes, finetuning data membership, and accuracy on downstream tasks.

## 1 Introduction

Finetuning of pretrained models such as Large Language Models (Devlin et al., ; Brown et al., ; Dubey et al., ) and Diffusion models (Ho et al. (), Rombach et al. ()) for improved performance on tasks such as generating images in a specific style or creating text for mathematical proofs has become an extremely common paradigm in deep learning. While full finetuning of all weights effectively boosts model capabilities, it requires a large amount of memory and computation time. In recent years, Low Rank Adapation (LoRA) (Hu et al., ), a method for finetuning where a learnable low rank decomposition is added to each weight matrix ($W_{i}\mapsto W_{i}+U_{i}V_{i}^{\top}$), has been used as an alternative to other finetuning methods thanks to its increased efficiency. LoRA finetuning and its variants have become widespread; there are now software packages (Mangrulkar et al., ; Han & Han, ), paid services (Replicate, ), and online communities (Civitai, ) in which countless LoRAs are trained and shared.

Given the ubiquity of LoRA weights, one can imagine treating them as a data type. As in recent works in the emerging field of weight-space learning, we can process the weights of input neural networks by using other neural networks (often termed metanetworks (Lim et al., ), neural functionals (Zhou et al., ), or deep weight space networks (Navon et al., )).

In this work, we are the first to extensively study applications, theory and architectures for Learning on LoRAs (LoL), which encompasses any tasks where LoRA weights are the input to some predictive model.
An LoL model acting on a LoRA could predict various useful properties of the underlying finetuned model. For instance, given LoRA weights of a finetuned model, an LoL model could predict the downstream accuracy of the model on some task, predict properties of the (potentially private) training data used to finetune the model, or edit the LoRA to work in some other setting. Recently, Salama et al. () and Dravid et al. () also consider learning tasks on LoRA weights, for some specific applications and a few types of models or learning algorithms.

When developing LoL architectures for processing LoRA weights, we account for the structure and symmetries of this unique data type. For one, the low rank decomposition $(U,V)$ generally has significantly fewer parameters ($(n+m)r$) than the dense matrix $UV^{\top}$ ($nm$). This can be leveraged to more efficiently learn tasks on LoRA weights. Moreover, for any invertible matrix $R\in\mathrm{GL}(r)$, we have that $URR^{-1}V^{\top}=UV^{\top}$, so the low rank decompositions $(U,V)$ and $(UR,VR^{-\top})$ are functionally equivalent. Because this transformation $\tau_{R}$ does not affect the underlying function represented by each LoRA, almost all relevant LoL problems are invariant to $\tau_{R}$. Therefore, the outputs of an effective model for any such task should be invariant to $\tau$.

Figure: Figure 1: Overview of Learning on LoRAs (LoL). A pretrained model $\theta_{\mathrm{base}}$ is finetuned to yield LoRA weight matrices $U_{1},V_{1},\ldots,U_{L},V_{L}$. These LoRA weights are taken as input to an LoL model $f_{\theta}$, which can make predictions such as the downstream accuracy of the finetuned model.
Refer to caption: x1.png

To this end, motivated by previous successes in invariant and equivariant weight space learning (Navon et al., ; Kofinas et al., ; Lim et al., ; Zhou et al., ; Kalogeropoulos et al., ) (and geometric deep learning in general (Bronstein et al., )), we propose several $\mathrm{GL}(r)$ equivariant and invariant neural architectures that can effectively process weights of LoRAs. Using various techniques from geometric deep learning such as canonicalization, invariant featurization, and equivariant linear maps, we develop several LoL models with various trade-offs in terms of efficiency, expressivity, and generalization.

To explore the feasibility of Learning on LoRAs tasks, and to analyze the effectiveness of our proposed architectures, we conduct experiments across various finetuned models. First, we create novel datasets for Learning on LoRAs; we train thousands of diverse LoRAs, which are finetuned from text-to-image diffusion generative models and language models. We then train LoL models on these LoRAs for prediction tasks such as: predict the CLIP score (Hessel et al., ) of a Stable Diffusion finetune given only its LoRA weights, predict attributes such as facial hair presence of the person that a diffusion model was personalized to, predict which data sources a language model LoRA was finetuned on, and predict the downstream reasoning accuracy of a language model LoRA. We find that simple $\mathrm{GL}(r)$-invariant LoL models can often perform some of these tasks well, and equivariant-layer based LoL models can do well across most tasks.

## 2 Background and Related Work

##### LoRA background and symmetries

Hu et al. () introduced the LoRA method, which is a parameter-efficient method for finetuning (typically large) models (Mangrulkar et al., ; Han et al., ). Consider a weight matrix $W\in\mathbb{R}^{n\times m}$ of some pretrained neural network. Directly finetuning $W$ would require training $nm$ parameters on additional data, which could be quite expensive. LoRA instead trains low-rank matrices $U\in\mathbb{R}^{n\times r}$ and $V\in\mathbb{R}^{m\times r}$ so that the new finetuned weight matrix is given by $W+UV^{\top}$. This only requires tuning $(n+m)r$ parameters, which is efficient as the rank $r$ is taken to be significantly lower than $n$ or $m$ (for instance a rank of $r=8$ can be sufficient to finetune a 7B-parameter language model with $n=m=4096$ (Liu et al., )).

As mentioned in the introduction, for any invertible matrix $R\in\mathrm{GL}(r)$, $W+(UR)(VR^{-\top})^{\top}=W+UV^{\top}$, so the LoRA update given by $(UR,VR^{-\top})$ is functionally equivalent to the one given by $(U,V)$. As a special case, when $Q\in\mathrm{O}(r)$ is orthogonal, $(UQ,VQ)$ is also functionally equivalent. Many variants of the original LoRA work have been proposed, and they often have comparable symmetries that can be handled similarly in our framework (Mangrulkar et al., ; Han et al., ); we discuss LoRA variants and their symmetries in Appendix E.

Prior works have also studied continuous symmetries (Thomas et al., ; Bogatskiy et al., ; Satorras et al., ; Lim et al., ; Pearce-Crump, ; Lawrence & Harris, ) such as rotation and scaling symmetries, especially for applications in the chemical and physical sciences. However, they study different classes of symmetries, such as $\mathrm{O}(n)$, $\mathrm{E}(n)$, and $\mathrm{SP}(n)$, none of which contain $\mathrm{GL}(n)$. To the best of our knowledge, we are the first to study $\mathrm{GL}(n)$ equivariant and invariant architectures.

##### Weight-space learning and metanetworks

Several neural network architectures have been developed that take in neural network weights as input (Unterthiner et al., ; Eilertsen et al., ; Metz et al., ; Peebles et al., ; Navon et al., ; ; Andreis et al., ; Shamsian et al., ). In many tasks, equivariance or invariance to parameter transformations that leave the network functionally unchanged has been found to be useful for empirical performance of weight-space networks (Navon et al., ; Zhou et al., ; Lim et al., ; Kalogeropoulos et al., ; Tran et al., ). However, these works are not specialized for LoRA weight spaces, since they consider different symmetry groups: many such works focus only on discrete permutation symmetries (Navon et al., ; Zhou et al., ; Lim et al., ; Zhou et al., ), or scaling symmetries induced by nonlinearities (Kalogeropoulos et al., ; Tran et al., ). As covered in the previous section, LoRA weights have general invertible symmetries, which include as special cases certain permutations and scaling symmetries between $U$ and $V^{\top}$.
While LoRAs contain inner permutation symmetries of the form $UP^{T}PV^{T}=UV^{\top}$, we emphasize that they generally do not have outer permutation symmetries of the more recognizable form $P_{1}UV^{T}P_{2}$, which would resemble those of standard weight spaces (e.g. MLPs or CNNs) more. This is an important difference; for instance, the outer permutation symmetries significantly harm linear merging of models trained from scratch (Entezari et al., ; Ainsworth et al., ; Lim et al., ), whereas finetuned models can be merged well with simple methods (Wortsman et al., ; Ilharco et al., ).

Recently, two works have explored the weight space of LoRA finetuned models for certain tasks. Salama et al. () develop an attack that, given the LoRA weights of a finetuned model, predicts the size of the dataset used to finetune the model. Inspired by correlations between finetuning dataset size and singular value magnitudes, they define a very specific type of LoL model that takes the singular values of dense, multiplied-out LoRA weights $\sigma_{1}(U_{i}V_{i}^{\top}),\ldots,\sigma_{r}(U_{i}V_{i}^{\top})$ as input. In contrast, we study more tasks, consider the problem of general LoL model design, and develop more expressive and efficient LoL models in our paper. In another context, Dravid et al. () study the LoRA weight space of finetuned personalized text-to-image diffusion models. They do not define LoL models, but instead study operations such as linear edits in the principal component space of their rank-one LoRA weights.

## 3 Learning on LoRAs Architectures

**Table 1: Properties of LoL architectures that we propose in this paper. Runtimes are for one LoRA layer with $U\in\mathbb{R}^{n\times r}$ and $V\in\mathbb{R}^{m\times r}$, so the rank $r$ is generally much lower than $n$ and $m$. Expressivity refers specifically to a notion of ability to fit $\mathrm{GL}$-invariant functions, which is formalized in Definition C.1.**
| Model | GL-Invariant | O-Invariant | Expressive | Preprocess Time | Forward Time |
| --- | --- | --- | --- | --- | --- |
| MLP | ✗ | ✗ | ✓ | $O((m+n)r)$ | $O((m+n)r)$ |
| MLP + $\mathrm{O}$-Align | ✗ | ✓ | ✓ | $O((m+n)r^{2})$ | $O((m+n)r)$ |
| MLP + SVD | ✓ | ✓ | ✗ | $O((m+n)r^{2})$ | $O((m+n)r)$ |
| MLP + Dense | ✓ | ✓ | ✓ | $O(mnr)$ | $O(mn)$ |
| $\mathrm{GL}$-net | ✓ | ✓ | ✓ | $O((m+n)r)$ | $O((m+n)r)$ |

In this section, we develop several neural network architectures for Learning on LoRAs. These have different trade-offs and properties, which we summarize in Table 1. First, we mathematically formalize the LoL learning problem. Then we describe four methods based on feeding features into a simple Multi-layer Perceptron (MLP). (^1^11These features can also be processed by other generic architectures, such as recurrent or Transformer-based sequence models along the layer dimension, but we use only MLPs in the work for simplicity.) Finally, we derive $\mathrm{GL}$-net, an architecture based on various equivariant and invariant modules.

### 3.1 Learning on LoRAs Models

We define a Learning on LoRAs (LoL) model that takes LoRA updates as input as follows. Suppose there are $L$ matrices in a base model that are finetuned via LoRA, i.e. the LoRA weights are $(U_{1},V_{1}),\ldots,(U_{L},V_{L})$, where $U_{i}\in\mathbb{R}^{n_{i}\times r}$ and $V_{i}\in\mathbb{R}^{m_{i}\times r}$. An LoL model with parameters $\theta$ and output space $\mathcal{Y}$ is a function $f_{\theta}:\mathbb{R}^{\sum_{i}n_{i}r+\sum_{i}m_{i}r}\to\mathcal{Y}$. An LoL model can output a scalar prediction ($\mathcal{Y}=\mathbb{R}$) or latent LoRA weight representations $(\tilde{U}_{1},\tilde{V}_{1}),\ldots,(\tilde{U}_{L},\tilde{V}_{L})$ where $\tilde{U}_{i}\in\mathbb{R}^{\tilde{n}_{i}\times r}$ and $\tilde{V}_{i}\in\mathbb{R}^{\tilde{m}_{i}\times r}$ ($\mathcal{Y}=\mathbb{R}^{\sum_{i}\tilde{n}_{i}r+\sum_{i}\tilde{m}_{i}r}$).

An LoL model is called *$\mathrm{GL}$-invariant* if for all $R_{1},\dots,R_{L}\in\mathrm{GL}(r)$

$$ $f_{\theta}(U_{1}R_{1},V_{1}R_{1}^{-\top},\ldots,U_{L}R_{L},V_{L}R_{L}^{-\top}) =f_{\theta}(U_{1},V_{1},\ldots,U_{L},V_{L}).$ (1) $$

This represents invariance to the action of the direct product $\mathrm{GL}(r)\times\cdots\times\mathrm{GL}(r)$ of $\mathrm{GL}(r)$ with itself $L$-times ($\mathrm{GL}(r)^{L}$). When the output space has decomposition structure, i.e. $f_{\theta}(U_{1},V_{1},\ldots,U_{L},V_{L})=((\tilde{U}_{1},\tilde{V}_{1}),
\ldots,(\tilde{U}_{L},\tilde{V}_{L}))$, we say that the model is $\mathrm{GL}(r)$-equivariant if for all $R_{1},\dots,R_{L}\in\mathrm{GL}(r)$,

$$ $f_{\theta}(U_{1}R_{1},V_{1}R_{1}^{-\top},\ldots,U_{L}R_{L},V_{L}R_{L}^{-\top}) =((\tilde{U}_{1}R_{1},\tilde{V}_{1}R_{1}^{-\top}),\ldots,(\tilde{U}_{L}R_{L}, \tilde{V}_{L}R_{L}^{-\top})).$ (2) $$

Expressivity is an important property of LoL models. Informally, an LoL model is universally expressive if it can fit any nice $\mathrm{GL}$-invariant function to arbitrary accuracy. We give a formal definition in Definition C.1, and prove our results in the context of this formal definition.

### 3.2 MLP-Based Methods with Featurization

##### MLP: Simple MLP on LoRA Weights

The simplest of our models is a simple MLP that takes the flattened LoRA weights as input: $\mathrm{MLP}_{\theta}(U_{1},V_{1},\ldots,U_{L},V_{L})$.
This method is not invariant to the LoRA parameter symmetries, but it is fully expressive, and it is efficient in that it does not need to form the $n$-by-$m$ dense matrices $UV^{\top}$.

##### MLP + O O \mathrm{O} roman_O -Align: Alignment for Canonicalization

One method for designing invariant or equivariant networks is to canonicalize the input by using a symmetry-invariant transformation to convert it into a canonical form (Kaba et al., ; Dym et al., ; Ma et al., ). For example, to canonicalize a dataset of point clouds with respect to rotations, one might choose one specific point cloud and then rotate all other point clouds in the dataset to it to maximize their relative similarity. After this alignment, any standard architecture can process the point clouds in an invariant manner without taking rotation into account. This kind of rotation alignment, known as $\mathrm{O}(r)$ canonicalization, admits a closed from solution and is significantly simpler than $\mathrm{GL}(r)$ canonicalization.

We canonicalize LoRA weights $U_{i},V_{i}$ into $\mathrm{O}(r)$-invariant representatives $U_{i}Q_{i},V_{i}Q_{i}$ as follows. First, we select template weights $\mathbf{U}_{i},\mathbf{V}_{i}$ of the same shape as $U_{i}$ and $V_{i}$ (in our experiments we randomly select $\mathbf{U}_{i}$ and $\mathbf{V}_{i}$ from our training set of LoRAs). Then we align $U_{i},V_{i}$ to the templates by finding the orthogonal matrix $Q_{i}$s that most closely match them in the Frobenius norm:

$$ $Q_{i}=\operatorname*{arg\,min}_{Q_{i}\in\mathrm{O}(r)}\left\lVert U_{i}Q_{i}- \mathbf{U}_{i}\right\rVert_{F}^{2}+\left\lVert V_{i}Q_{i}-\mathbf{V}_{i}\right \rVert_{F}^{2}.$ (3) $$

This is an instance of the well-known Orthogonal Procrustes problem (Schönemann, ), which has a closed form solution: $Q_{i}$ can be computed from the singular value decomposition of the $r$-by-$r$ matrix $U_{i}^{\top}\mathbf{U}_{i}+V_{i}^{\top}\mathbf{V}_{i}$, which can be computed efficiently when the rank $r$ is low. After computing these $Q_{i}s$ for pair of LoRA weights, we input the $U_{i}Q_{i},V_{i}Q_{i}$ into an MLP to compute predictions.

##### MLP + SVD: Singular Values as Features

Similarly to Salama et al. (), we also consider an LoL architecture that feeds the singular values of the multiplied-out LoRA weights, $\sigma_{1}(U_{i}V_{i}^{\top}),\ldots,\sigma_{r}(U_{i}V_{i}^{\top})$ into an MLP to compute predictions (though Salama et al. () mostly use nearest neighbor predictors instead of an MLP). Since $U_{i}V_{i}^{\top}$ is rank $r$, there are at most $r$ nonzero singular values. These singular values are $\mathrm{GL}$-invariant features, but they are not expressive. For instance, negating $U_{i}$ does not affect the singular values, but it can completely change the functionality and destroy the performance of the finetuned model.

##### MLP + Dense: Multiplying-Out LoRA Weights

Lastly, a simple, natural method is to first perform matrix multiplications to compute the full dense matrices $U_{i}V_{i}^{\top}$, and then apply a machine learning model to these dense matrices. In this paper, we will flatten and concatenate these dense matrices, and then apply a simple MLP to them. This approach is fully $\mathrm{GL}$-invariant, and is universally expressive, but it is computationally expensive since the dense matrices are of size $nm$ as opposed to the size $(n+m)r$ of the low rank decomposition.

### 3.3 GL GL \mathrm{GL} roman_GL -net: Constructing GL-invariant Models using Equivariant Layers

Figure: Figure 2: Architecture of $\mathrm{GL}$-net. Blue boxes are equivariant representations, and red boxes are invariant representations. First, equivariant linear maps lower the dimension of the input. Then our $\mathrm{GL}$ equivariant nonlinearities and more equivariant linear maps process the features. Finally, a matrix multiplication head computes invariant features that are processed by an MLP.
Refer to caption: x2.png

A common and effective method for parameterizing invariant neural networks is to first process the input with equivariant layers and then make the final prediction with invariant modules (Cohen & Welling, ; Maron et al., ; ; Bronstein et al., ). Thus, we develop $\mathrm{GL}$-net, which consists of a series of GL-equivariant linear layers and equivariant nonlinearities followed by a GL-invariant head to predict invariant characteristics of the input. See Figure 2 for an illustration.

#### 3.3.1 Equivariant Linear Layers

For LoRA weights $U_{i}\in\mathbb{R}^{n_{i}\times r}$ and $V_{i}\in\mathbb{R}^{m_{i}\times r}$, we describe the $\mathrm{GL}$-equivariant linear layers, which map to new weights $\tilde{U}_{i}\in\mathbb{R}^{n_{i}^{\prime}\times r}$ and $\tilde{V}_{i}\in\mathbb{R}^{m_{i}^{\prime}\times r}$. For each of these LoRA weights, we have LoL model parameters $\mathbf{\Phi}_{i}\in\mathbb{R}^{n_{i}^{\prime}\times n_{i}}$ and $\mathbf{\Psi}_{i}\in\mathbb{R}^{m_{i}^{\prime}\times m_{i}}$. The equivariant linear map is given by:

$$ $F_{\mathrm{Linear}}\left(U_{1},V_{1},\ldots,U_{L},V_{L}\right)=\left(\mathbf{ \Phi}_{1}U_{1},\mathbf{\Psi}_{1}V_{1},\ldots,\mathbf{\Phi}_{L}U_{L},\mathbf{ \Psi}_{L}V_{L}\right)$ (4) $$

That is, a $\mathrm{GL}$ equivariant linear layer consists of left matrix multiplying each $U_{i}$ by a learnable $\mathbf{\Phi}_{i}$, and left matrix multiplying each $V_{i}$ by a learnable $\mathbf{\Psi}_{i}$. In practice, we choose the same hidden dimension for every $\tilde{U}_{i}$ and $\tilde{V}_{i}$ for simplicity, so $n_{1}^{\prime}=m_{1}^{\prime}=\ldots=n_{L}^{\prime}=m_{L}^{\prime}$. The equivariance of $F_{\mathrm{Linear}}$ can be easily seen. For instance, if there is only one layer ($L=1$), we have

$$ $F_{\mathrm{Linear}}(U_{1}R,V_{1}R^{-\top})=(\mathbf{\Phi}_{1}U_{1}R,\mathbf{ \Psi}_{1}V_{1}R^{-\top})=(\tilde{U}_{1}R,\tilde{V}_{1}R^{-\top})=R\star F_{ \mathrm{Linear}}(U_{1},V_{1}),$ (5) $$

where $(\tilde{U}_{1},\tilde{V}_{1})=F_{\mathrm{Linear}}(U_{1},V_{1})$, and $R\star(\tilde{U}_{1},\tilde{V}_{1})=(\tilde{U}_{1}R,\tilde{V}_{1}R^{-\top})$ is the action of $R$ on the LoRA space. In fact, we can show a stronger statement — these linear maps in equation 4 constitute *all possible* $\mathrm{GL}$-equivariant linear maps. See Appendix A for the proof.

###### Proposition 1 .

All linear $\mathrm{GL}$-equivariant layers can be written in the form of equation 4.

###### Proposition 1 .

##### Extension to Convolution LoRAs.

Low rank convolution decompositions are generally represented as two consecutive convolutions where $C_{B}$ projects the input down from $m$ to $r$ channels and $C_{A}$ projects the input up to $n$. For our architecture we flatten all dimensions of the convolutions except the hidden channel dimension, and then we apply equivariant linear layers.

#### 3.3.2 Equivariant Non-linearity

Equivariant networks often interleave pointwise non-linearities with linear equivariant layers. Unfortunately, non-trivial pointwise non-linearities are not equivariant to our symmetry group. A general recipe for designing equivariant non-linearities is taking $f_{\mathrm{equi}}(\mathbf{x})=f_{\mathrm{inv}}(\mathbf{x})\cdot\mathbf{x}$, for some scalar invariant function $f_{\mathrm{inv}}$ (Thomas et al., ; Villar et al., ; Blum-Smith & Villar, ). In our case, given any non-linearity $\sigma:\mathbb{R}\to\mathbb{R}$, we define a $\mathrm{GL}$-equivariant non-linearity by

$$ $\sigma_{\mathrm{GL}}(U)_{i}=\sigma\Big{(}\sum\nolimits_{j}(UV^{\top})_{ij}\Big {)}U_{i},\qquad\sigma_{\mathrm{GL}}(V)_{i}=\sigma\Big{(}\sum\nolimits_{j}(UV^{ \top})_{ji}\Big{)}V_{i}.$ (6) $$

In other words, we scale the $i$-th row of $U_{i}$ by a quantity that depends on the the $i$-th row sum of $UV^{\top}$, and scale the $i$-th row of $V_{i}$ by a quantity that depends on the $i$-th column sum of $UV^{\top}$.
Since $UV^{\top}$ is invariant under the action of $\mathrm{GL}$, and $\mathrm{GL}$ acts independently on each row of $U$ and $V$, $\sigma_{\mathrm{GL}}$ is equivariant (we prove this in Appendix B).
In our experiments, we often take $\sigma(x)=\mathrm{ReLU}(\mathrm{sign}(x))$, which has the effect of zero-ing out entire rows of $U$ or $V$ in a $\mathrm{GL}$-equivariant way — this is a natural $\mathrm{GL}$-equivariant generalization of ReLU.
For our experiments we tune the number of equivariant linear layers and we frequently find that only one is required, so we often do not use this equivariant non-linearity; nonetheless, it may be more useful in other applications, such as $\mathrm{GL}$-equivariant tasks.

#### 3.3.3 Invariant Head

Many equivariant architectures used for classification use an invariant aggregation step before applying a standard classifier model. For invariant classification tasks, we use an invariant head which consists of the relatively simple operation of computing the LoRA matrix products, concatenating them, and then applying an MLP on top. Explicitly, this is given as $f_{\mathrm{inv}}(U_{1},V_{1},\ldots,U_{L},V_{L})=\mathrm{MLP}(\mathrm{cat}[U_{
1}V_{1}^{\top},\ldots,U_{L}V_{L}^{\top}])$, where $\mathrm{cat}$ concatenates the entries of the inputs into a flattened vector.
For the input, computing $U_{i}V_{i}^{T}$ would be both memory and time expensive. To avoid this, we use equivariant linear layers to lower the dimension of $U_{i},V_{i}$ to about $32\times r$, such that $U_{i}V_{i}^{\top}\in\mathbb{R}^{32\times 32}$ is efficient to compute – see Figure 2 for an illustration.

## 4 Theory

Here, we restate the properties of our models as described in Section 3 and Table 1. All proofs of these results are in the Appendix.

###### Theorem 1 (Invariance) .

MLP + Dense, MLP + SVD, $\mathrm{GL}$-net are $\mathrm{GL}$-Invariant. MLP + $\mathrm{O}$-Align is $\mathrm{O}$-Invariant but not $\mathrm{GL}$-Invariant. MLP is not $\mathrm{GL}$ or $\mathrm{O}$-Invariant.

###### Theorem 2 (Universality) .

MLP, MLP + $\mathrm{O}$-Align, MLP + Dense, and $\mathrm{GL}$-net can arbitrarily approximate any $GL$-invariant continuous function on a compact set of full rank matrices.

###### Theorem 1 (Invariance) .

###### Theorem 2 (Universality) .

## 5 Experimental Results

**Table 2: We report train and test mean squared error (MSE) for predicting normalized CLIP score of CelebA-LoRA models. We also show test Kendall’s $\tau$ and $R^{2}$ coefficients. $\mathrm{GL}$-net has significantly lower test MSE than any other LoL model, whereas a standard MLP does no better than random guessing. MLPs with $\mathrm{O}$-Align, SVD, or Dense featurization all perform similarly on the test set.**
| LoL Model | Train MSE | Test MSE | $\tau$ | $R^{2}$ |
| --- | --- | --- | --- | --- |
| MLP | $.047\pm.004$ | $.988\pm.005$ | $.226\pm.016$ | $.333\pm.022$ |
| MLP + $\mathrm{O}$-Align | $.001\pm.001$ | $.175\pm.006$ | $.654\pm.004$ | $.856\pm.004$ |
| MLP + SVD | $.071\pm.003$ | $.148\pm.005$ | $.667\pm.008$ | $.863\pm.006$ |
| MLP + Dense | $.013\pm.002$ | $.169\pm.011$ | $.677\pm.006$ | $.871\pm.005$ |
| $\mathrm{GL}$-net | $.009\pm.001$ | $\mathbf{.111}\pm.004$ | $\mathbf{.695}\pm.005$ | $\mathbf{.884}\pm.003$ |

In this section, we present experimental results evaluating our five LoL models across tasks involving finetuned diffusion and language models (Subsections  5.2- 5.3). Our experiments demonstrate the efficacy of LoL models in solving GL-invariant tasks on LoRAs. Additionally, we explore these models’ ability to generalize to LoRAs of previously unseen ranks, with detailed findings presented in Subsection 5.4.

### 5.1 Datasets of Trained LoRA Weights

We generate three new datasets of LoRAs of varying ranks. Two of these datasets correspond to finetunes of diffusion models on image datasets, while the last corresponds to finetunes of a language model on text. Our datasets have diversity in terms of LoRA rank, training hyperparameters (two datasets have randomly sampled hyperparameters, whereas one has fixed hyperparameters), and base model architecture.

##### CelebA-LoRA.

We train a dataset of 3,900 LoRA finetuned Stable Diffusion 1.4 models (Rombach et al., ) using the PEFT library (Mangrulkar et al., ). Each LoRA is rank 4, and is finetuned via DreamBooth personalization (Ruiz et al., ) on 21 images of a given celebrity in the CelebA dataset (Liu et al., ). Further, each LoRA is trained with randomly sampled hyperparameters (gradient accumulation steps, train steps, learning rate, prompt) and initialization. We train LoL models to predict hyperparameters, CLIP scores, and training images of finetuned diffusion models by using their LoRA weights.

##### Imagenette-LoRA.

We also use Dreambooth to finetune another 2,046 Stable Diffusion 1.4 models with LoRA rank 32 on different subsets of the Imagenette dataset (Howard, ) — a subset of ImageNet (Deng et al., ) consisting of images from ten dissimilar classes. Unlike CelebA LoRA, we use the same training hyperparameters for each finetuning run (but vary initialization, random seed, and finetuning dataset).

##### Qwen2-ARC-LoRA.

We create a dataset of 2,000 language model LoRAs by finetuning the 1.5 billion parameter Qwen2 model (Yang et al., ) on subsets of the training set of the commonly used ARC dataset (Clark et al., ), which is a dataset for testing question answering and science knowledge. The ARC dataset consists of questions from many data sources. For each LoRA, we randomly sample a subset of 19 data sources, and omit data from the unsampled data sources. Also, each LoRA is randomly initialized and trained with randomly sampled hyperparameters. See Appendix D.2 for more details.

### 5.2 Diffusion Model Classification

#### 5.2.1 CLIP Score prediction.

CLIP scores were introduced by Hessel et al. () as an automated method for measuring text-image-alignment in diffusion models by evaluating the semantic similarity between their prompts and generated images. Higher CLIP scores generally correspond to better diffusion models, so CLIP score is a useful metric for evaluating these models. We calculate the CLIP score of each of our 3,900 diffusion models (with 33 fixed prompts) and train LoL models on the task of determining the CLIP score of a model given its finetuned weights; see Appendix D.1.1 for more details.

The results are shown in Table 2. All of the LoL models (besides the vanilla MLP) can effectively predict the CLIP score of diffusion LoRAs using only their decomposed low rank matrices. The poor performance of vanilla MLP demonstrates the importance of $\mathrm{GL}$ symmetries, whereas the relatively good performance of MLP + Dense suggests “outer” permutation symmetries are less important for LoL models, confirming two of our hypotheses from Section 2. $\mathrm{GL}$-net is able to calculate CLIP score significantly faster than generating images, which requires 660 score-network forward passes per finetuned model. Other baselines, including MLP + SVD, MLP + $\mathrm{O}$-Align and MLP + Dense are less predictive. To demonstrate the utility of this task, we generate images from the two models in our test set predicted by $\mathrm{GL}$-net to have the highest and lowest CLIP scores respectively. As shown in Figure 3, $\mathrm{GL}$-net’s predictions are highly correlated with the actual quality of model output.

Figure: Figure 3: Images generated by two diffusion models in our test set. (a), (c), and (e) correspond to images generated by the model predicted by $\mathrm{GL}$-net to have the highest CLIP score. (b), (d), and (f) correspond to outputs of the model predicted by $\mathrm{GL}$-net to have the lowest CLIP score.
Refer to caption: extracted/5926522/figures/good_images/img_3.jpg

**Table 3: Results for using LoL models to predict properties of the finetuning data of diffusion models, given only the LoRA weights. In the left section, the task is to predict 5 different binary attributes of the CelebA celebrity that each LoRA was finetuned on. In the right section, the task is to predict which of the 10 Imagenette classes were included in the finetuning data of the LoRA. MLP and MLP + SVD have trouble fitting these tasks, while $\mathrm{GL}$-net performs the best overall.**
|  | CelebA Attributes | Imagenette Classes |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| LoL Model | Train Loss | Test Loss | Test Acc | Train Loss | Test Loss | Test Acc |
| MLP | $.551\pm.000$ | $.554\pm.000$ | $72.4\pm 0.0$ | $.582\pm.004$ | $.709\pm.004$ | $49.6\pm 1.3$ |
| MLP + $\mathrm{O}$-Align | $.074\pm.022$ | $.333\pm.008$ | $87.2\pm 0.5$ | $.002\pm.001$ | $.278\pm.008$ | $87.8\pm 0.3$ |
| MLP + SVD | $.490\pm.015$ | $.509\pm.013$ | $77.3\pm 1.3$ | $.581\pm.013$ | $.638\pm.013$ | $65.6\pm 0.6$ |
| MLP + Dense | $.131\pm.018$ | $.267\pm.007$ | $89.1\pm 0.4$ | $.019\pm.001$ | $.264\pm.011$ | $88.9\pm 0.6$ |
| $\mathrm{GL}$-net | $.064\pm.008$ | $\textbf{.232}\pm\textbf{.007}$ | $\textbf{91.3}\pm\textbf{0.1}$ | $.019\pm.000$ | $\textbf{.244}\pm\textbf{.005}$ | $\textbf{90.4}\pm\textbf{0.3}$ |

#### 5.2.2 Predicting Properties of Training Data

##### Dataset attribute prediction.

On our CelebA-LoRA dataset, we train LoL models to classify attributes of the celebrity that each LoRA was finetuned on. Due to the noisy nature of CelebA labels, we follow the advice of (Lingenfelter et al., ) and only train and test on the five CelebA attributes they determine to be least noisy. Also, we train LoL models to predict which Imagenette classes appeared in the finetuning data of each model in Imagenette-LoRA. Our results are in Table 3.

##### Dataset size prediction.

The task of predicting the size of the finetuning dataset given LoRA weights was first studied recently by Salama et al. (). The success of this task implies a privacy leak, since model developers may sometimes wish to keep the size of their finetuning dataset private. Moreover, the dataset size is a useful quantity to know for data membership inference attacks and model inversion attacks, so accurately predicting dataset size could improve effectiveness of these attacks too (Shokri et al., ; Haim et al., ).

**Table 4: Results for finetuning dataset size prediction with LoL models. We use our Imagenette-LoRA dataset, and predict the number of classes (or equivalently images) that are used to finetune each model.**
| LoL Model | Train Loss | Test Acc |
| --- | --- | --- |
| MLP | $.439\pm.030$ | $20.9\pm 3.1$ |
| MLP + $\mathrm{O}$-Align | $.014\pm.004$ | $35.2\pm 2.1$ |
| MLP + SVD | $.082\pm.010$ | $\mathbf{73.1}\pm 1.6$ |
| MLP + Dense | $.382\pm.119$ | $31.6\pm 4.3$ |
| $\mathrm{GL}$-net | $.005\pm.003$ | $46.8\pm 2.1$ |

In Table 4, we show results for finetuning-dataset-size prediction using LoL models. The task is to take in the LoRA weights of one of the diffusion models from our Imagenette-LoRA dataset, and predict the number of unique images that it was finetuned on. Although it struggles on some other tasks, we see that MLP + SVD is able to effectively predict dataset size, in line with observations from Salama et al. (). Other LoL models struggle to generalize well on this task. Interestingly, $\mathrm{GL}$-net performs approximately as well as expected if using the following strategy: first predict which classes are present in the finetuning set of the LoRA (as in Table 3), and then sum the number of classes present to predict the dataset size. MLP+SVD is clearly using different predictive strategies for this task, as it cannot predict which individual classes are present (Table 3), but it can predict the number of classes present. See Appendix D.3 for more analysis on the learned prediction strategies of the LoL models on this task.

### 5.3 Language Model Classification

**Table 5: LoL model performance on LoRAs of the Qwen2 1.5B language model. The left column is prediction of which data sources the input LoRA was finetuned on, the middle column is prediction of the validation loss for the finetuning task, and the right column is prediction of the accuracy of the LoRA on the ARC-C (Clark et al., ) test set. All metrics are reported on the LoL task’s test set (on held-out input networks). Higher numbers are better on all metrics.**
|  | LoL Model Prediction Target |  |  |
| --- | --- | --- | --- |
| LoL Model | Data Membership (Acc) | Val Loss ($R^{2}$) | ARC-C Acc ($R^{2}$) |
| MLP | $.516\pm.006$ | $.113\pm.059$ | $.107\pm.035$ |
| MLP + $\mathrm{O}$-Align | $.550\pm.016$ | $.821\pm.078$ | $.965\pm.004$ |
| MLP + SVD | $.551\pm.001$ | $\mathbf{.999}\pm.000$ | $.983\pm.002$ |
| MLP + Dense | $\mathbf{.625}\pm.008$ | $.987\pm.003$ | $.981\pm.002$ |
| $\mathrm{GL}$-net | $.605\pm.007$ | $.998\pm.000$ | $\mathbf{.987}\pm.001$ |

In this section, we experiment with LoL models on our Qwen2-ARC-LoRA dataset of finetuned language models. For our first task, we consider a type of data membership inference task: we aim to predict whether each of the 19 data sources was used to train a given LoRA (this is a 19-label binary classification task). We also consider two performance prediction task: we train LoL models to predict, for each LoRA, the validation loss on the finetuning objective, and the downstream accuracy on the ARC-C test set (Clark et al., ).

Results are in Table 5. Several of our LoL models are very successful at predicting the finetuning validation loss and ARC-C test accuracy of LoRA-finetuned language models, with some of them achieving .99 $R^{2}$ on finetuning validation loss regression and over .98 $R^{2}$ on ARC-C test accuracy regression. This could be useful in evaluations of finetuned models, as standard evaluations of language models can require a lot of time and resources, whereas our LoL models can evaluate these models with one quick forward pass. However, data membership inference is a harder task for the LoL models, with the best model achieving only 62.5% test accuracy in predicting which data sources were present in the finetuning dataset. Though our setup is quite different, these results could be related to work showing that model-based membership inference attacks are challenging for LLMs (Duan et al., ; Das et al., ).

### 5.4 Generalization to Unseen Ranks

Figure: Figure 4: Performance of LoL models across inputs of varying ranks. Each model is only trained on rank $4$ LoRA weights from CelebA-LoRAs. (Left) Test accuracy on CelebA attribute prediction. (Right) Test loss on CelebA attribute prediction. MLP + Dense and $\mathrm{GL}$-net generalize well to ranks that are unseen during training, but do face degradation at rank one. On the other hand, MLP + SVD does not generalize well.
Refer to caption: x3.png

In the previous sections, each LoL model was trained on LoRA weights of one fixed rank (e.g. rank 4 for the LLMs, rank 32 for the Imagenette models). In practice, model developers can choose different LoRA ranks for finetuning their models, even when they are finetuning the same base model for similar tasks. Thus, we may desire LoL models that are effective for inputs of different ranks. In this section, we explore an even more difficult problem — whether LoL models can generalize to ranks that are unseen during training time.

The MLP + Dense and $\mathrm{GL}$-net models can directly take as input models of different ranks. We also use MLP + SVD on inputs of different ranks, by parameterizing the model for inputs of rank $r$, then truncating to top $r$ singular values for inputs of rank greater than $r$, and zero-padding to length $r$ for inputs of rank less than $r$.

In Figure 4, we train LoL models on inputs of rank $4$, and then test performance on inputs of ranks between $1$ and $32$. On the CelebA attribute prediction task, we see that MLP + Dense and $\mathrm{GL}$-net mostly generalize very well to different ranks that are unseen during training (except not as well for $r=1$), while MLP + SVD does not generalize as well.

### 5.5 Runtime and Scaling to Large Models

Figure: Figure 5: (Left) data preprocessing time for LoL models across 64 inputs of varying sizes. (Right) forward pass time for LoL models across 512 inputs of varying sizes. MLP + Dense runs out of memory for largest inputs.
Refer to caption: x4.png

Previous weight-space models generally take in small networks as inputs, e.g. 5,000 parameters (Unterthiner et al., ), 80,000 parameters (Lim et al., ), or sometimes up to 4 million parameters (Navon et al., ). However, many of the most impactful neural networks have orders of magnitude more parameters. Thus, here we consider the runtime and scalability of LoL models as we increase the model size. We will consider all of the language model sizes in the GPT-3 family (Brown et al., ), which range from 125M to 175B parameters. We assume that we finetune rank 4 LoRA weights for one attention parameter matrix for each layer.

In Figure 5, we show the time it takes for data preprocessing of 64 input networks, and forward passes for 512 input networks (with batch size up to 64). Even at the largest scales, it only takes at most seconds to preprocess and compute LoL model forward passes for each input LoRA. Every LoL model besides MLP + Dense scales well with the model size: forward passes barely take longer at larger model sizes, and data preprocessing is still limited to less than a second per input network.

## 6 Conclusion

In this work, we introduced the Learning on LoRAs framework, and investigated architectures, theory, and applications for it. There are many potential applications and future directions for using LoL models to process finetunes. For instance, future work could explore equivariant tasks, such as those that involve editing or merging LoRAs. Additionally, future work could consider learning an LoL model that can generalize across different model architectures, or different base models of the same architecture.

### Acknowledgements

DL is supported by an NSF Graduate Fellowship. YG is supported by the the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Autonomous and Intelligent Machines and Systems (grant reference EP/S024050/1). SJ is supported by NSF Award CCF-2112665 (TILOS AI Institute), NSF Award 2134108 and an MIT Systems that Learn award. HM is the Robert J. Shillman Fellow, and is supported by the Israel Science Foundation through a personal grant (ISF 264/23) and an equipment grant (ISF 532/23).

## Appendix A GL GL \mathrm{GL} roman_GL Equivariant Linear Maps Characterization

Here, we characterize the form of $\mathrm{GL}$-equivariant linear maps, and provide the proof of Proposition 1. We use basic representation theory techniques, which are similar to the ones used to characterize equivariant linear maps in the geometric deep learning literature (Maron et al., ; Finzi et al., ; Navon et al., ).

### A.1 Proof of Proposition 1

First, we prove a lemma, that allows us to analyze equivariant linear maps between direct sums of spaces in terms of a direct sum of equivariant linear maps between the constituent spaces. For a group $G$, we denote the vector space of $G$-equivariant linear maps from $\mathcal{V}$ to $\mathcal{W}$ by $\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})$. This is closely related to a result from Navon et al. () that characterizes the matrices underlying linear maps between direct sums.

###### Lemma 1 .

Let $G$ be a group and let $\mathcal{V}_{1},\dots,\mathcal{V}_{n}$, $\mathcal{W}_{1},\dots,\mathcal{W}_{m}$ be $G$-representations. If $\mathcal{V}=\mathcal{V}_{1}\oplus\cdots\oplus\mathcal{V}_{n}$ and $\mathcal{W}=\mathcal{W}_{1}\oplus\cdots\oplus\mathcal{W}_{m}$ are direct sums of representations then

$$ $\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})\cong\bigoplus_{i,j}\mathrm{Hom}_{G}( \mathcal{V}_{i},\mathcal{W}_{j}).$ (7) $$

###### Proof.

We construct an explicit isomorphism $F:\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})\to\bigoplus_{i,j}\mathrm{Hom}_{G}(
\mathcal{V}_{i},\mathcal{W}_{j})$. Let $\mathrm{inc}_{i}:\mathcal{V}_{i}\to\mathcal{V}$ denote the inclusion of $\mathcal{V}_{i}$ in $\mathcal{V}$, and $\mathrm{proj}_{j}:\mathcal{W}\to\mathcal{W}_{j}$ denote the projection from $\mathcal{W}$ to $\mathcal{W}_{j}$. For $\phi\in\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})$, define:

$$ $F(\phi)=(\mathrm{proj}_{j}\circ\phi\circ\mathrm{inc}_{i})_{i,j}$ (8) $$

$F$ is clearly linear, being defined by composition of linear maps. Moreover, since $\phi$, $\mathrm{inc}_{i}$, and $\mathrm{proj}_{j}$ are all $G$-equivariant, their composition $\phi_{i,j}=\mathrm{proj}_{j}\circ\phi\circ\mathrm{inc}_{i}$ is also $G$-equivariant. To prove injectivity, suppose $F(\phi)=F(\psi)$ for some $\phi,\psi\in\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})$. Then $\forall v=(v_{1},\dots,v_{n})\in\mathcal{V}$

$$ $\displaystyle\phi(v)$ $\displaystyle=(\mathrm{proj}_{1}(\phi(v)),\dots,\mathrm{proj}_{m}(\phi(v)))$ $\displaystyle=\left(\sum_{i}\mathrm{proj}_{1}(\phi(\mathrm{inc}_{i}(v_{i}))), \dots,\sum_{i}\mathrm{proj}_{m}(\phi(\mathrm{inc}_{i}(v_{i})))\right)$ $\displaystyle=\left(\sum_{i}\mathrm{proj}_{1}(\psi(\mathrm{inc}_{i}(v_{i}))), \dots,\sum_{i}\mathrm{proj}_{m}(\psi(\mathrm{inc}_{i}(v_{i})))\right)$ $\displaystyle=\psi(v).$ $$

For surjectivity, given $(\phi_{i,j})_{i,j}\in\bigoplus_{i,j}\mathrm{Hom}_{G}(\mathcal{V}_{i},\mathcal{
W}_{j})$, define $\phi:\mathcal{V}\to\mathcal{W}$ by

$$ $\phi(v_{1},\dots,v_{n})=\left(\sum_{i}\phi_{i,1}(v_{i}),\dots,\sum_{i}\phi_{i, m}(v_{i})\right).$ (9) $$

$\phi$ is linear by construction; to show $G$-equivariance, let $g\in G$ and $(v_{1},\dots,v_{n})\in\mathcal{V}$

$$ $\displaystyle\phi(g\cdot(v_{1},\dots,v_{n}))$ $\displaystyle=\phi(g\cdot v_{1},\dots,g\cdot v_{n})$ $\displaystyle=\left(\sum_{i}\phi_{i,1}(g\cdot v_{i}),\dots,\sum_{i}\phi_{i,m}( g\cdot v_{i})\right)$ $\displaystyle=\left(\sum_{i}g\cdot\phi_{i,1}(v_{i}),\dots,\sum_{i}g\cdot\phi_{ i,m}(v_{i})\right)$ $\displaystyle=g\cdot\phi(v_{1},\dots,v_{n}),$ $$

where we use the $G$-equivariance of each $\phi_{i,j}$. Thus, $\phi\in\mathrm{Hom}_{G}(\mathcal{V},\mathcal{W})$, and by construction $F(\phi)=(\phi_{i,j})_{i,j}$. Therefore, $F$ is a linear bijection.
∎

###### Proof of Proposition 1 .

Let $\mathcal{I}:=({\mathcal{U}}_{1}\oplus{\mathcal{V}}_{1}^{*})\oplus\cdots\oplus(
{\mathcal{U}}_{L}\oplus{\mathcal{V}}_{L}^{*})$ and $\mathcal{O}:=(\tilde{{\mathcal{U}}}_{1}\oplus\tilde{{\mathcal{V}}}_{1}^{*})
\oplus\cdots\oplus(\tilde{{\mathcal{U}}}_{L}\oplus\tilde{{\mathcal{V}}}_{L}^{*})$ be the input and output spaces of an equivariant LoL model. Denote $\dim(\mathcal{U}_{i})=n_{i}\cdot r$, $\dim(\mathcal{V}_{i}^{*})=r\cdot m_{i}$, $\dim(\tilde{\mathcal{U}}_{i})=n_{i}^{\prime}\cdot r$, $\dim(\tilde{\mathcal{V}}_{i}^{*})=r\cdot m_{i}^{\prime}$, and $G:=\mathrm{GL}(r)^{L}=\mathrm{GL}(r)\times\cdots\times\mathrm{GL}(r)$. We are interested in characterizing the vector space of $G$-equivariant linear maps, which we denote by $\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O})$. In the main text we show that the maps $F_{\mathrm{Linear}}^{\mathbf{\Phi}_{1},\mathbf{\Psi}_{1},\ldots,\mathbf{\Phi}_
{L},\mathbf{\Psi}_{L}}$, defined by

$$ $F_{\mathrm{Linear}}^{\mathbf{\Phi}_{1},\mathbf{\Psi}_{1},\ldots,\mathbf{\Phi}_ {L},\mathbf{\Psi}_{L}}\left(U_{1},V_{1},\ldots,U_{L},V_{L}\right)=\left( \mathbf{\Phi}_{1}U_{1},\mathbf{\Psi}_{1}V_{1},\ldots,\mathbf{\Phi}_{L}U_{L}, \mathbf{\Psi}_{L}V_{L}\right)$ (10) $$

are all $G$-equivariant. In other words, we showed that

$$ $\mathcal{L}:=\left\{F_{\mathrm{Linear}}^{\mathbf{\Phi}_{1},\mathbf{\Psi}_{1}, \ldots,\mathbf{\Phi}_{L},\mathbf{\Psi}_{L}}\mid\mathbf{\Phi}_{i}\in\mathbb{R}^ {n_{i}^{\prime}\times n_{i}},\mathbf{\Psi}_{i}\in\mathbb{R}^{m_{i}^{\prime} \times m_{i}}\right\}\subseteq\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O}).$ (11) $$

$\mathcal{L}$ is a linear subspace of dimension $\sum_{i=1}^{L}n_{i}n_{i}^{\prime}+\sum_{i=1}^{L}m_{i}m_{i}^{\prime}$, so in order to prove that $\mathcal{L}=\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O})$, it’s enough to show that $\dim(\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O}))=\sum_{i=1}^{L}n_{i}n_{i}^{
\prime}+\sum_{i=1}^{L}m_{i}m_{i}^{\prime}$. Since $\mathcal{I}$ and $\mathcal{O}$ are direct sums of representations, Lemma 1 implies that the dimension of $\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O})$ is the sum of the dimensions of the constituents:

$$ $\begin{split}\dim(\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O}))=\sum_{i=1}^{L} \sum_{i^{\prime}=1}^{L}\biggl{(}\dim\left(\mathrm{Hom}_{G}(\mathcal{U}_{i}, \tilde{\mathcal{U}}_{i^{\prime}})\right)+\dim\left(\mathrm{Hom}_{G}(\mathcal{U }_{i},\tilde{\mathcal{V}}_{i^{\prime}}^{*})\right)+\\ \dim\left(\mathrm{Hom}_{G}(\mathcal{V}_{i}^{*},\tilde{\mathcal{U}}_{i^{\prime} })\right)+\dim\left(\mathrm{Hom}_{G}(\mathcal{V}_{i}^{*},\tilde{\mathcal{V}}_{ i^{\prime}}^{*})\right)\biggr{)}.\end{split}$ (12) $$

Next, note that $\mathcal{U}_{i}$, $\mathcal{V}_{i}^{*}$, $\tilde{\mathcal{U}}_{i}$, and $\tilde{\mathcal{V}}_{i}^{*}$ all decompose into irreducible representations

$$ $\mathcal{U}_{i}=\bigoplus_{j=1}^{n_{i}}\mathcal{U}_{i}^{j},\,\mathcal{V}_{i}^{ *}=\bigoplus_{j=1}^{m_{i}}(\mathcal{V}_{i}^{j})^{*},\,\tilde{\mathcal{U}}_{i}= \bigoplus_{j=1}^{n_{i}^{\prime}}\tilde{\mathcal{U}}_{i}^{j},\,\tilde{\mathcal{ V}}_{i}^{*}=\bigoplus_{j=1}^{m_{i}^{\prime}}(\tilde{\mathcal{V}}_{i}^{j})^{*},$ (13) $$

each of which is isomorphic to the standard representation of the $i$-th copy of $\mathrm{GL}(r)$ on either $\mathbb{R}^{r}$ (for $\mathcal{U}_{i}^{j}$ and $\tilde{\mathcal{U}}_{i}^{j}$) or $(\mathbb{R}^{r})^{*}$ (for $(\mathcal{V}_{i}^{j})^{*}$ and $(\tilde{\mathcal{V}}_{i}^{j})^{*}$). We can think of $\mathcal{U}_{i}^{j}$ as the space spanned by the $j$-th row in the input $U_{i}$ matrix and $(\mathcal{V}_{i}^{j})^{*}$ as the space spanned by the $j$-th column of the input $V_{i}^{\top}$ matrix. This decomposition gives us

$$ $\begin{split}\dim\left(\mathrm{Hom}_{G}(\mathcal{U}_{i},\tilde{\mathcal{U}}_{i ^{\prime}})\right)&=\sum_{j=1}^{n_{i}}\sum_{j^{\prime}=1}^{n_{i}^{\prime}}\dim \left(\mathrm{Hom}_{G}(\mathcal{U}_{i}^{j},\tilde{\mathcal{U}}_{i^{\prime}}^{j ^{\prime}})\right),\\ \dim\left(\mathrm{Hom}_{G}(\mathcal{U}_{i},\tilde{\mathcal{V}}_{i^{\prime}}^{* })\right)&=\sum_{j=1}^{n_{i}}\sum_{j^{\prime}=1}^{m_{i}^{\prime}}\dim\left( \mathrm{Hom}_{G}(\mathcal{U}_{i}^{j},(\tilde{\mathcal{V}}_{i^{\prime}}^{j^{ \prime}})^{*})\right),\\ \dim\left(\mathrm{Hom}_{G}(\mathcal{V}_{i}^{*},\tilde{\mathcal{U}}_{i^{\prime} })\right)&=\sum_{j=1}^{m_{i}}\sum_{j^{\prime}=1}^{n_{i}^{\prime}}\dim\left( \mathrm{Hom}_{G}((\mathcal{V}_{i}^{j})^{*},\tilde{\mathcal{U}}_{i^{\prime}}^{j ^{\prime}})\right),\\ \dim\left(\mathrm{Hom}_{G}(\mathcal{V}_{i}^{*},\tilde{\mathcal{V}}_{i^{\prime} }^{*})\right)&=\sum_{j=1}^{m_{i}}\sum_{j^{\prime}=1}^{m_{i}^{\prime}}\dim\left (\mathrm{Hom}_{G}((\mathcal{V}_{i}^{j})^{*},(\tilde{\mathcal{V}}_{i^{\prime}}^ {j^{\prime}})^{*})\right).\end{split}$ (14) $$

Since these are all irreducible representations, Schur’s lemma (Fulton & Harris, ) implies that the dimension of the space of $G$-equivariant maps is either $1$ (if the representations are isomorphic) or 0 0 (if they are not). Notice that,

- •
$\mathcal{U}_{i}^{j}$ and $\tilde{\mathcal{U}}_{i^{\prime}}^{j^{\prime}}$ are isomorphic as $G$-representations if and only if $i=i^{\prime}$ (if $i\neq i^{\prime}$ different copies of $\mathrm{GL}(r)$ act on $\mathcal{U}_{i}^{j}$ and $\tilde{\mathcal{U}}_{i^{\prime}}^{j^{\prime}}$).
- •
$(\mathcal{V}_{i}^{j})^{*}$ and $(\tilde{\mathcal{V}}_{i^{\prime}}^{j^{\prime}})^{*}$ are isomorphic as $G$-representations if and only if $i=i^{\prime}$ (if $i\neq i^{\prime}$ different copies of $\mathrm{GL}(r)$ act on $(\mathcal{V}_{i}^{j})^{*}$ and $(\tilde{\mathcal{V}}_{i^{\prime}}^{j^{\prime}})^{*}$).
- •
$\mathcal{U}_{i}^{j}$ is never isomorphic to $(\tilde{\mathcal{V}}_{i^{\prime}}^{j^{\prime}})^{*}$ and $(\mathcal{V}_{i}^{j})^{*}$ is never isomorphic to $\tilde{\mathcal{U}}_{i^{\prime}}^{j^{\prime}}$.

Therefore, together with Equation 12 and Equation 14 we get

$$ $\begin{split}\dim\left(\mathrm{Hom}_{G}(\mathcal{I},\mathcal{O})\right)&=\sum_ {i=1}^{L}\sum_{i^{\prime}=1}^{L}\left(n_{i}n_{i^{\prime}}^{\prime}\cdot\mathbf {1}_{i=i^{\prime}}+m_{i}m_{i^{\prime}}^{\prime}\cdot\mathbf{1}_{i=i^{\prime}} \right)\\ &=\sum_{i=1}^{L}n_{i}n_{i}^{\prime}+\sum_{i=1}^{L}m_{i}m_{i}^{\prime}\end{ split},$ (15) $$

concluding the proof.
∎

###### Lemma 1 .

###### Proof.

###### Proof of Proposition 1 .

## Appendix B Proof of Invariances: Theorem 1

We prove invariance (or lack thereof) of each model one by one. For this section, consider arbitrary inputs $\mathbf{UV}=(U_{1},V_{1},\ldots,U_{L},V_{l})$, where $U_{i}\in\mathbb{R}^{n_{i}\times r}$ and $V_{i}\times\mathbb{R}^{m_{i}\times r}$. Further, choose invertible $\mathbf{R}=(R_{1},\ldots,R_{L})\in\mathrm{GL}(r)^{L}$. Denote the application of $\mathbf{R}$ on $\mathbf{UV}$ by $\mathbf{R}\star\mathbf{UV}=(U_{1}R_{1},V_{1}R_{1}^{-\top},\ldots,U_{L}R_{L},V_
{L}R_{L}^{-\top})$ To show that a function $f$ is $\mathrm{GL}$ invariant, we will show that $f(\mathbf{R}\star\mathbf{UV})=f(\mathbf{UV})$. Note that, since $\mathrm{O}(r)\subset\mathrm{GL}(r)$, if we show that a function is $\mathrm{GL}$-invariant, then it is also $\mathrm{O}$-invariant.

##### MLP.

We will show that the simple MLP is neither $\mathrm{O}$-invariant or $\mathrm{GL}$-invariant. It suffices to show that it is not $\mathrm{O}$-invariant. As a simple example, let $\mathrm{MLP}(U,V)=U_{1,1}$ output the top-left entry of $U$. Then let $U$ be a matrix with $1$ in the top-left corner and 0 0 elsewhere. Further, let $P$ be the permutation matrix that swaps the first and second entries of its input. Then $\mathrm{MLP}(UP,VP)=0\neq\mathrm{MLP}(U,V)=1$. As permutation matrices are orthogonal, $P\in\mathrm{O}(r)$, so the MLP is indeed not $\mathrm{O}(r)$ invariant.

##### MLP + O O \mathrm{O} roman_O -Align.

We will show that the $\mathrm{O}$-alignment approach is $\mathrm{O}$-invariant on all but a Lebesgue-measure-zero set. For simplicity, let $L=1$, $U\in\mathbb{R}^{n\times r}$, and $V\in\mathbb{R}^{m\times r}$. Let $\mathbf{U}\in\mathbb{R}^{n\times r}$ and $\mathbf{V}\in\mathbb{R}^{m\times r}$ be the template matrices, which we assume are full rank (the full rank matrices are a Lebesgue-dense set, so this is an allowed assumption). Recall that we canonicalize $U,V$ as $\rho(U,V)=(UQ,VQ)$, where:

$$ $Q=\operatorname*{arg\,min}_{Q\in O(r)}\left\lVert UQ-\mathbf{U}\right\rVert_{F }^{2}+\left\lVert VQ-\mathbf{V}\right\rVert_{F}^{2}.$ (16) $$

We call any solution to this problem a *canonicalizing matrix* for $(U,V)$. This can be equivalently written as

$$ $Q=\operatorname*{arg\,min}_{Q\in O(r)}\left\lVert\begin{bmatrix}U\\ V\end{bmatrix}Q-\begin{bmatrix}\mathbf{U}\\ \mathbf{V}\end{bmatrix}\right\rVert_{F}^{2}.$ (17) $$

Let $M=U^{\top}\mathbf{U}+V^{\top}\mathbf{V}$. Then a global minimum of this problem is $Q=AB^{\top}$, where $A\Sigma B^{\top}$ is an SVD of $M$. If $M$ has distinct singular values, then $A$ and $B$ are unique up to sign flips, and $AB^{\top}$ is in fact unique. Thus, if $M$ has unique singular values, $(UQ,VQ)$ is unique. We assume from here that $M$ has distinct singular values, as the set of all $M$ that do form a Lebesgue-dense subset of $\mathbb{R}^{r\times r}$ (and thus this is satisfies for Lebesgue-almost-every $U$ and $V$).

Now, to show $\mathrm{O}$-invariance, let $\tilde{Q}\in\mathrm{O}(r)$. We will show that $\rho(U\tilde{Q},V\tilde{Q})=\rho(U,V)$.

$$ $\operatorname*{arg\,min}_{Q\in O(r)}\left\lVert\begin{bmatrix}U\tilde{Q}\\ V\tilde{Q}\end{bmatrix}Q-\begin{bmatrix}\mathbf{U}\\ \mathbf{V}\end{bmatrix}\right\rVert_{F}^{2}=\operatorname*{arg\,min}_{Q^{ \prime}\in O(r)}\left\lVert\begin{bmatrix}U\\ V\end{bmatrix}Q^{\prime}-\begin{bmatrix}\mathbf{U}\\ \mathbf{V}\end{bmatrix}\right\rVert_{F}^{2}$ (18) $$

because the orthogonal matrices are closed under multiplication. Thus, if $Q$ is a canonicalizing matrix for $(U,V)$, then $\tilde{Q}^{\top}Q$ is a canonicalizing matrix for $(U\tilde{Q},V\tilde{Q})$. Moreover, we have that $\tilde{Q}^{\top}U^{\top}\mathbf{U}+\tilde{Q}^{\top}V^{\top}\mathbf{V}=\tilde{Q
}^{\top}M$, so this has the same singular values as $M$ (as orthogonal matrices don’t affect singular values); this means that the singular values are distinct, so there is a unique canonicalizing matrix for $(U\tilde{Q},V\tilde{Q})$. This means that the canonicalization matrix must be equal to $\tilde{Q}^{\top}Q$, so that

$$ $\rho(U\tilde{Q},V\tilde{Q})=(U\tilde{Q}\tilde{Q}^{\top}Q,V\tilde{Q}\tilde{Q}^{ \top}Q)=(UQ,VQ)=\rho(U,V).$ (19) $$

As this argument holds for Lebesgue-almost-every $U$ and $V$, we have shown $\mathrm{O}$-invariance of MLP + $\mathrm{O}$-Align.

Finally, we have to show that this LoL model is not $\mathrm{GL}$-invariant. To do this, let the MLP approximate the Frobenius norm function on its first input, so $\mathrm{MLP}(U,V)\approx\left\lVert U\right\rVert$. Consider any $U$ that is nonzero, and let $a>2$ be a scalar. Then $aI\in\mathrm{GL}(r)$. Also, we have that $\rho(U,V)=(UQ,VQ)$ for some $Q\in\mathrm{O}(r)$, so this canonicalization does not affect the Frobenius norm of the first entry. Thus, we have that

$$ $\mathrm{MLP}(\rho(U,V))\approx\left\lVert U\right\rVert\neq a\left\lVert U \right\rVert\approx\mathrm{MLP}(\rho(aU,(1/a)V)).$ (20) $$

So MLP + $\mathrm{O}$-Align is not invariant under $\mathrm{GL}$.

##### MLP + SVD.

We show this is $\mathrm{GL}$-invariant. The output of this model $f$ on $\mathbf{UV}$ can be written as

$$ $\mathrm{MLP}(\sigma_{1}(U_{1}V_{1}^{\top}),\ldots,\sigma_{r}(U_{1}V_{1}^{\top} ),\ldots,\sigma_{1}(U_{L}V_{L}^{\top}),\ldots,\sigma_{r}(U_{L}V_{L}^{\top}).$ (21) $$

Where $\sigma_{j}(U_{i}V_{i}^{\top})$ is the $j$th singular value of $U_{i}V_{i}^{\top}$. Thus, we can compute that:

$$ $\displaystyle f(\mathbf{R}\star\mathbf{UV})$ $\displaystyle=\mathrm{MLP}(\sigma_{1}(U_{1}R_{1}R_{1}^{-1}V_{1}^{\top}),\ldots ,\sigma_{r}(U_{L}R_{L}R_{L}^{-1}V_{L}^{\top}))$ (22) $\displaystyle=\mathrm{MLP}(\sigma_{1}(U_{1}V_{1}^{\top}),\ldots,\sigma_{r}(U_{ L}V_{L}^{\top}))$ (23) $\displaystyle=f(\mathbf{UV}).$ (24) $$

##### MLP + Dense.

We show this is $\mathrm{GL}$-invariant. We can simply see that

$$ $\displaystyle\mathrm{MLP}(\mathbf{R}\star\mathbf{UV})$ $\displaystyle=\mathrm{MLP}(U_{1}R_{1}R_{1}^{-1}V_{1}^{\top},\ldots,U_{L}R_{L}R _{L}^{-1}V_{L}^{\top})$ (25) $\displaystyle=\mathrm{MLP}(U_{1}V_{1}^{\top},\ldots,U_{L}V_{L}^{\top})$ (26) $\displaystyle=\mathrm{MLP}(\mathbf{UV}).$ (27) $$

##### GL GL \mathrm{GL} roman_GL -net.

We show this is $\mathrm{GL}$-invariant. First, assume that the equivariant linear layers are $\mathrm{GL}$-equivariant, and the equivariant nonlinearities are $\mathrm{GL}$-equivariant. Note that the invariant head of $\mathrm{GL}$-net is a special case of MLP + Dense, which we have already proven to be invariant. Further, an invariant function composed with an equivariant function is invariant, so we are done.

We only need to prove that the nonlinearities are $\mathrm{GL}$-equivariant, because we have already proven that the equivariant linear layers are $\mathrm{GL}$-equivariant in the main text. Let $\sigma:\mathbb{R}\to\mathbb{R}$ be any real function, and recall that the $\mathrm{GL}$-equivariant nonlinearity takes the following form on $U\in\mathbb{R}^{n\times r}$, $V\in\mathbb{R}^{m\times r}$:

$$ $\sigma_{\mathrm{GL}}(U,V)=(\tilde{U},\tilde{V}),\qquad\tilde{U}_{i}=\sigma\Big {(}\sum\nolimits_{j}(UV^{\top})_{ij}\Big{)}U_{i},\qquad\tilde{V}_{i}=\sigma \Big{(}\sum\nolimits_{j}(UV^{\top})_{ji}\Big{)}V_{i},$ (28) $$

where $\tilde{U}\in\mathbb{R}^{n\times r}$, $\tilde{V}\in\mathbb{R}^{m\times r}$, and for example $U_{i}\in\mathbb{R}^{r}$ denotes the $i$th row of $U_{i}$.

###### Lemma 2 .

For any real function $\sigma:\mathbb{R}\to\mathbb{R}$, the function $\sigma_{\mathrm{GL}}$ defined in equation 28 is $\mathrm{GL}$ equivariant.

###### Proof.

Let $U\in\mathbb{R}^{n\times r},V\in\mathbb{R}^{m\times r},$ and $R\in\mathrm{GL}(r)$ be arbitrary. Denote the output of the nonlinearity on the transformed weights as $\sigma_{\mathrm{GL}}(UR,VR^{-\top})=(\tilde{U}^{(R)},\tilde{V}^{(R)})$. Then we have that

$$ $\displaystyle\tilde{U}_{i}^{(R)}$ $\displaystyle=\sigma\Big{(}\sum_{j}(URR^{-1}V^{\top})_{ij}\Big{)}R^{\top}U_{i} =R^{\top}\left[\sigma\Big{(}\sum_{j}(UV^{\top})_{ij}\Big{)}U_{i}\right]=R^{ \top}U_{i}$ (29) $\displaystyle\tilde{V}_{i}^{(R)}$ $\displaystyle=\sigma\Big{(}\sum_{j}(URR^{-1}V^{\top})_{ji}\Big{)}R^{-1}V_{i}=R ^{-1}\left[\sigma\Big{(}\sum_{j}(UV^{\top})_{ji}\Big{)}V_{i}\right]=R^{-1}V_{i}.$ (30) $$

In matrix form, this means that $\tilde{U}_{i}^{(R)}=\tilde{U}_{i}R$ and $\tilde{V}_{i}^{(R)}=\tilde{V}_{i}R^{-\top}$. In other words,

$$ $\sigma_{\mathrm{GL}}(UR,VR^{-1})=R\star\sigma_{\mathrm{GL}}(U,V),$ (31) $$

which is the definition of $\mathrm{GL}$-equivariance, so we are done.
∎

###### Lemma 2 .

###### Proof.

## Appendix C Proof of Expressivity: Theorem 2

In this section we formally restate and prove Theorem 2. We start by defining full rank $\mathrm{GL}$ universality for LoL models.

###### Definition C.1 (Full rank $\mathrm{GL}$ -universality) .

Let $\mathcal{D}=\{(U_{1},V_{1},\dots,U_{L},V_{L})\mid U_{i}\in\mathbb{R}^{n_{i}
\times r},V_{i}\in\mathbb{R}^{m_{i}},\mathrm{rank}(U_{i})=\mathrm{rank}(V_{i})
=r\}$ be the set of LoRA updates of full rank. A LoL architecture is called full rank $\mathrm{GL}$-universal if for every $\mathrm{GL}$-invariant function $f:\mathcal{D}\to\mathbb{R}$, every $\epsilon>0$, and every compact set $K\subset\mathcal{D}$, there is a model $f^{\mathrm{LoL}}$ of said architecture that approximates $f$ on $K$ up to $\epsilon$:

$$ $\sup_{{\bm{X}}\in K}|f^{\mathrm{LoL}}({\bm{X}})-f({\bm{X}})\rvert<\epsilon.$ (32) $$

Note that the set $\mathcal{D}$ of full-rank LoRA updates is Lebesgue-dense (its compliment has measure 0 0).

###### Theorem 3 (Formal restatement of Theorem 2 ) .

The MLP, MLP + $\mathrm{O}$-Align, MLP + Dense, and $\mathrm{GL}$-net LoL architectures are all full rank $\mathrm{GL}$ universal.

The universality of MLP and MLP + O-Align models follows from the universal approximation thereom for MLPs (Hornik et al., ). To prove Theorem 3 for MLP + Dense and $\mathrm{GL}$-net we use the following result from Dym & Gortler ().

###### Proposition 2 (Proposition 1.3 from Dym & Gortler ( 2024 ) ) .

Let $\mathcal{M}$ be a topological space, and $G$ a group which acts on $\mathcal{M}$. Let $K\subset\mathcal{M}$ be a compact set, and let $f^{\mathrm{inv}}:\mathcal{M}\to\mathbb{R}^{N}$ be a continuous $G$-invariant map that separates orbits. Then for every continuous invariant function $f:\mathcal{M}\to\mathbb{R}$ there exists some continuous $f^{\mathrm{general}}:\mathbb{R}^{N}\to\mathbb{R}$ such that $f(x)=f^{\mathrm{general}}(f^{\mathrm{inv}}(x))$, $\forall x\in K$.

In order to use the proposition above for MLP + Dense LoL models, we need to show that multiplying out the LoRA updates separates $\mathrm{GL}$ orbits.

###### Lemma 3 .

The function

$$ $f^{\mathrm{mul}}(U_{1},V_{1},\dots,U_{L},V_{L})=(U_{1}V_{1}^{\top},\dots,U_{L} V_{L}^{\top}),$ $$

separates $\mathrm{GL}$-orbits in $\mathcal{D}$. That is, if $(U_{1},V_{1},\dots,U_{L},V_{L})$ and $(U_{1}^{\prime},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$ are in different $G$-orbits then $f^{\mathrm{mul}}(U_{1},V_{1},\dots,U_{L},V_{L})\neq f^{\mathrm{mul}}(U_{1}^{
\prime},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$.

###### Proof.

We prove the contrapositive. Let $(U_{1},V_{1},\dots,U_{L},V_{L})$ and $(U_{1}^{\prime},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$ be LoRA updates such that $f^{\mathrm{mul}}(U_{1},V_{1},\dots,U_{L},V_{L})=f^{\mathrm{mul}}(U_{1}^{\prime
},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$, i.e. $\forall i\in\{1,\dots,L\}$

$$ $U_{i}V_{i}^{\top}=U_{i}^{\prime}V_{i}^{\prime\top}.$ (33) $$

Since $U_{i}$, $V_{i}$, $U_{i}^{\prime}$, and $V_{i}^{\prime}$ are of rank $r$, their corresponding Gram matrices $U_{i}^{\top}U_{i}$, $V_{i}^{\top}V_{i}$, $U_{i}^{\prime\top}U_{i}^{\prime},V_{i}^{\prime\top}V_{i}^{\prime}\in\mathbb{R}
^{r\times r}$, are also of rank $r$ and are thus invertible. Multiplying both sides of Equation 33 by $V_{i}(V_{i}^{\top}V_{i})^{-1}$ from the right, we get

$$ $U_{i}\cancel{V_{i}^{\top}V_{i}(V_{i}^{\top}V_{i})^{-1}}=U_{i}^{\prime} \underbrace{V_{i}^{\prime\top}V_{i}(V_{i}^{\top}V_{i})^{-1}}_{R_{i}}.$ (34) $$

Substituting $U_{i}^{\prime}R_{i}$ back to Equation 33 we get

$$ $U_{i}^{\prime}R_{i}V_{i}^{\top}=U_{i}^{\prime}V_{i}^{\prime\top}.$ $$

Multiplying by $(U_{i}^{\prime\top}U_{i}^{\prime})^{-1}U_{i}^{\prime\top}$ from the left gives

$$ $\cancel{(U_{i}^{\prime\top}U_{i}^{\prime})^{-1}U_{i}^{\prime\top}U_{i}^{\prime }}R_{i}V_{i}^{\top}=\cancel{(U_{i}^{\prime\top}U_{i}^{\prime})^{-1}U_{i}^{ \prime\top}U_{i}^{\prime}}V_{i}^{\prime\top}.$ $$

Therefore, to prove $(U_{1},V_{1},\dots,U_{L},V_{L})$ and $(U_{1}^{\prime},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$ are in the same orbit all we need to do is show that $R_{i}\in\mathrm{GL}(r)$. To do so it’s enough to show that $V_{i}^{\prime\top}V_{i}$ is invertible. And indeed, starting from Equation33

$$ $\displaystyle U_{i}V_{i}^{\top}$ $\displaystyle=U_{i}^{\prime}V_{i}^{\prime\top}$ (35) (Multiply both sides by $(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}$ from the left) $\displaystyle(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i}V_{i}^{\top}$ $\displaystyle=(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i}^{\prime}V_{i}^{\prime\top}$ (Simplify left side: $(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i}=I$ ) $\displaystyle V_{i}^{\top}$ $\displaystyle=(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i}^{\prime}V_{i}^{\prime\top}$ (Multiply both sides by $V_{i}$ from the right) $\displaystyle V_{i}^{\top}V_{i}$ $\displaystyle=(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i}^{\prime}V_{i}^{\prime \top}V_{i}$ (Multiply both sides by $(V_{i}^{\top}V_{i})^{-1}$ from the left) $\displaystyle I$ $\displaystyle=(V_{i}^{\top}V_{i})^{-1}(U_{i}^{\top}U_{i})^{-1}U_{i}^{\top}U_{i }^{\prime}V_{i}^{\prime\top}V_{i}.$ $$

Therefore, $R_{1},\dots,R_{L}\in\mathrm{GL}(r)$, $U_{i}=U_{i}^{\prime}R$, and $V_{i}=V_{i}^{\prime}R_{i}^{-\top}$, implying that $(U_{1},V_{1},\dots,U_{L},V_{L})$ and $(U_{1}^{\prime},V_{1}^{\prime},\dots,U_{L}^{\prime},V_{L}^{\prime})$ are in the same $G$-orbit.
∎

We are now ready to prove Theorem 3.

###### Proof of theorem 3 .

Let $f:\mathcal{D}\to\mathbb{R}$ be a continuous $\mathrm{GL}$-invariant function, let $K\subset\mathcal{D}$ be a compact set and fix $\epsilon>0$.

- 1.
MLP. Since $K$ is compact and $f$ is continuous, universality follows from the universal approximation theorem for MLPs (Hornik et al., ).
- 2.
MLP + O-Align. Let $f^{\mathrm{align}}$ be the O-canonicalization function. $f^{\mathrm{align}}$ is continuous so $f^{\mathrm{align}}(K)$ is compact. and let $f^{\mathrm{MLP}}$ be an MLP that approximates $f$ up-to $\epsilon$ on $f^{\mathrm{align}}(K)$.
$\begin{split}\sup_{{\bm{X}}\in K}\lvert f^{\mathrm{MLP}}(f^{\mathrm{align}}({
\bm{X}}))-f({\bm{X}})\rvert&=\sup_{{\bm{X}}\in K}\lvert f^{\mathrm{MLP}}(f^{
\mathrm{align}}({\bm{X}}))-f(f^{\mathrm{align}}({\bm{X}}))\rvert\\
&=\sup_{{\bm{Y}}\in f^{\mathrm{align}}(K)}\lvert f^{\mathrm{MLP}}({\bm{Y}})-f(
{\bm{Y}})\rvert<\epsilon.\end{split}$
The first equality holds since $f$ is $\mathrm{GL}$-invariant, and in particular $\mathrm{O}$-invariant.
- 3.
MLP + Dense. From Lemma 3 we know that $f^{\mathrm{mul}}$ separates orbits. It’s additionally clear that $f^{\mathrm{mul}}$ is continuous and $G$-invariant. Therefore, using Proposition 2 there exists a function $f^{\mathrm{general}}:\mathbb{R}^{N}\to\mathbb{R}$ such that $f\equiv f^{\mathrm{general}}\circ f^{\mathrm{mul}}$ on $K$. Since $f^{\mathrm{mul}}$ is continuous, $f^{\mathrm{mul}}(K)$ is also compact and we can use the universal approximation theorem of MLPs for $f^{\mathrm{genral}}$ on $f^{\mathrm{mul}}(K)$. Therefore, there exists an MLP $f^{\mathrm{MLP}}$ such that
$\begin{split}\sup_{{\bm{X}}\in K}\lvert f^{\mathrm{MLP}}(f^{\mathrm{mul}}({\bm
{X}}))-f({\bm{X}})\rvert&=\sup_{{\bm{X}}\in K}\lvert f^{\mathrm{MLP}}(f^{
\mathrm{mul}}({\bm{X}}))-f^{\mathrm{general}}(f^{\mathrm{mul}}({\bm{X}}))
\rvert\\
&=\sup_{{\bm{Y}}\in f^{\mathrm{mul}}(K)}\lvert f^{\mathrm{MLP}}({\bm{Y}})-f^{
\mathrm{general}}({\bm{Y}})\rvert<\epsilon\end{split}$
(36)
- 4.
GL-net. Since we proved MLP + Dense is universal, it’s enough to show that $\mathrm{GL}$-net can implement MLP + Dense. If we take a $\mathrm{GL}$-net with no equivariant layers (or equivalently a single equivariant layer that implements the identity by setting $\mathbf{\Phi}_{i}=I_{n_{i}}$, $\mathbf{\Psi}_{i}=I_{m_{i}}$) and apply the invariant head directly to the input, the resulting model is exactly MLP + Dense.

∎

###### Definition C.1 (Full rank GL GL \mathrm{GL} roman_GL -universality) .

###### Theorem 3 (Formal restatement of Theorem 2 ) .

###### Proposition 2 (Proposition 1.3 from Dym & Gortler ( 2024 ) ) .

###### Lemma 3 .

###### Proof.

###### Proof of theorem 3 .

## Appendix D Experimental Details

### D.1 Diffusion Model LoRA Datasets

#### D.1.1 CelebA Finetuning

**Table 6: Hyperparameter distributions for the 3,900 different LoRA diffusion model finetunes we trained. $U(S)$ denotes the uniform distribution over a set $S$.**
| Hyperparameter | Distribution |
| --- | --- |
| Learning rate | $U(\{10^{-4},3\cdot 10^{-4},10^{-3},3\cdot 10^{-3}\})$ |
| Train Steps | $U(\{100,133,167,200\})$ |
| Batch Size | $U(\{1,2\})$ |
| Prompt | $U(\{\text{``Celebrity"},\text{``Person"},\text{``thing"},\text{``skd"}\})$ |
| Rank | 4 |

We finetune 3,900 models on various celebrities in the CelebA dataset (Liu et al., ) using the DreamBooth personalization method (Ruiz et al., ). Each LoRA is personalized to one celebrity by finetuning on 21 images of that celebrity. Every LoRA is trained starting from a different random initialization, with hyperparameters randomly sampled from reasonable distributions — see Table 6 for the distributions.

##### CLIP score prediction.

For CLIP score prediction, we use the following prompts, the first thirty of which are from PartiPrompts Yu et al. (), and the last three of which are written to include words relevant to prompts the models are finetuned on. We use a fixed random seed of 42 for image generation. For predicting CLIP scores, $\mathrm{GL}$-net takes the mean of each matrix product $U_{i}V_{i}^{\top}$ in the invariant head. This is equivalent to using an equivariant hidden dimension of $1$.

> 1. ’a red sphere on top of a yellow box’, 2. ’a chimpanzee sitting on a wooden bench’, 3. ’a clock tower’, 4. ’toy cars’, 5. ’a white rabbit in blue jogging clothes doubled over in pain while a turtle wearing a red tank top dashes confidently through the finish line’, 6. ’a train going to the moon’, 7. ’Four cats surrounding a dog’, 8. ’the Eiffel Tower in a desert’, 9. ’The Millennium Wheel next to the Statue of Liberty. The Sagrada Familia church is also visible.’, 10. ’A punk rock squirrel in a studded leather jacket shouting into a microphone while standing on a stump and holding a beer on dark stage.’, 11. ’The Statue of Liberty surrounded by helicopters’, 12. ’A television made of water that displays an image of a cityscape at night.’, 13. ’a family on a road trip’, 14. ’the mona lisa wearing a cowboy hat and screaming a punk song into a microphone’, 15. ’a family of four posing at Mount Rushmore’, 16. ’force’, 17. ’an oil surrealist painting of a dreamworld on a seashore where clocks and watches appear to be inexplicably limp and melting in the desolate landscape. a table on the left, with a golden watch swarmed by ants. a strange fleshy creature in the center of the painting’, 18. ’Downtown Austin at sunrise. detailed ink wash.’, 19. ’A helicopter flies over Yosemite.’, 20. ’A giraffe walking through a green grass covered field’, 21. ’a corgi’s head’, 22. ’portrait of a well-dressed raccoon, oil painting in the style of Rembrandt’, 23. ’a volcano’, 24. ’happiness’, 25. ”the words ’KEEP OFF THE GRASS’ on a black sticker”, 26. ’A heart made of wood’, 27. ’a pixel art corgi pizza’, 28. ’two wine bottles’, 29. ’A funny Rube Goldberg machine made out of metal’, 30. ’a horned owl with a graduation cap and diploma’, 31. ’A celebrity in a park’, 32. ’A person on the beach’, 33. ’A thing in a city’

##### CelebA Attribute Prediction.

As mentioned in Section 5.2.2, we only predict the five least noisy CelebA attributes, as measured by Lingenfelter et al. (). Recall that each LoRA in CelebA-LoRA is trained on 21 images of a given celebrity. The attribute labels can vary across these 21 images, so we say the ground truth attribute label of the LoRA is the majority label across these 21 images.

#### D.1.2 ImageNette Finetuning

For Imagenette (Howard, ), we finetune 2,046 models. In particular, for each nonempty subset of the 10 Imagenette classes, we finetune 2 models using 1 image from each present class. We use a rank of 32, in part so that we can test how well LoL models perform on larger ranks than the other experiments.

**Table 7: Hyperparameters for the dataset of 2,046 different LoRA diffusion model finetunes we trained on Imagenette (Imagenette-LoRA).**
| Hyperparameter | Value |
| --- | --- |
| Learning rate | $3\cdot 10^{-4}$ |
| Train Steps | $150$ |
| Batch Size | $1$ |
| Prompt | $sks_photo$ |
| Rank | 32 |

### D.2 Language Model LoRA Dataset

Here, we describe the details of our Qwen2-ARC-LoRA dataset of trained language model LoRAs.

For training data, we use the ARC training set (Clark et al., ), using both the easy and challenge splits. First, we hold out a fixed validation set sampled from this set to compute the validation loss on. Each training data point originates from one of 20 sources (e.g. some questions come from Ohio Achievement Tests, and some come from the Virginia Standards of Learning). For each LoRA finetuning run, we sample a random subset of these sources to use as training data (except we do not ever omit the Mercury source, which contains many more data points than the other sources). To do this, we sample a random integer in $s\in[1,19]$, then choose a random size-$s$ subset of the 19 possibly-filtered sources to drop.

**Table 8: Hyperparameter distributions for the 2,000 different LoRA language model finetunes we trained (Qwen2-ARC-LoRA). $U(S)$ denotes the uniform distribution over a set $S$.**
| Hyperparameter | Distribution |
| --- | --- |
| Learning rate | $10^{U([-5,-3])}$ |
| Weight decay | $10^{U([-6,-2])}$ |
| Epochs | $U(\{2,3,4\})$ |
| Batch Size | $U(\{32,64,128\})$ |
| LoRA Dropout | $U([0,.1])$ |
| Filtered sources | $U(\text{sources})$ |

For each LoRA finetuning run, we sample random hyperparameters: learning rate, weight decay, number of epochs, batch size, LoRA dropout, and the data sources to filter. The distributions from which we sample are shown in Table 8, and were chosen to give reasonable but varied performance across different runs. All trained LoRAs were of rank 4, and we only applied LoRA to tune the key and value projection matrices of each language model.

### D.3 Dataset size prediction

Here we describe theoretical and experimental details relating to the implicit strategies that $\mathrm{GL}$-net and MLP + SVD may employ in dataset size prediction as in Table 4.

Suppose that a model $f_{\mathrm{class}}$ trained to predict Imagenette class presence of LoRAs (as in Table 3) has test accuracy $p$, while each of the 10 classes is present with probability $.5$. For LoRA weights $x$, $f_{\mathrm{class}}(x)_{i}=1$ if the model predicts that class $i$ is in the finetuning dataset of $x$. Define the corresponding dataset-size prediction model $f_{\mathrm{size}}(x)=\sum_{i=1}^{10}f_{\mathrm{class}}(x)_{i}$ That is, $f_{\mathrm{size}}$ predicts whether each class is in the dataset, sums up these predictions, and then outputs the sum as the expected dataset size.

Assuming $f_{\mathrm{class}}$ is equally likely to provide false-negative or false-positive predictions for each class, each of its 10 outputs has probability $1-p$ of being incorrect. Consider an input LoRA $x$ with a dataset size $s$. Then $f_{\mathrm{size}}(x)=\hat{y}_{1}+\hat{y}_{2}+\dots\hat{y}_{10}$, where $\hat{y}_{i}$ is equal to $y_{i}$ with probability $p$, and $1-y_{i}$ with probability $1-p$. So,

$$ $f_{\mathrm{size}}(x)=s\iff|\{i\ |\ y_{i}=1\land\hat{y}_{i}=0\}|=|\{i\ |\ y_{i} =0\land\hat{y}_{i}=1\}|$ (37) $$

The cardinality of the left set is distributed as $\mathrm{binom}(s,1-p)$, while the cardinality of the right set is distributed as $\mathrm{binom}(10-s,1-p)$. So, $\mathbb{P}(f_{\mathrm{size}}(x)=s|s)=\mathbb{P}(\beta_{1}=\beta_{2})$, for $\beta_{1}\sim\mathrm{binom}(s,1-p)$ and $\beta_{2}\sim\mathrm{binom}(10-s,1-p)$. Thus,

$$ $\mathbb{P}_{(x,s)\sim\mathrm{data}}(f_{\mathrm{size}}(x)=s)=\sum_{\eta=0}^{10} [\mathbb{P}\left(\mathrm{binom}(\eta,1-p)=\mathrm{binom}(10-\eta,1-p)\ |\ \eta \right)\cdot\mathbb{P}(\eta)]$ (38) $$

Table 3 shows that for $\mathrm{GL}$-net, $f_{\mathrm{class}}$ is correct with probability $p=.904$. So, we have $1-p=.096$.

Using a Python script to evaluate equation 38, we find that the probability that $f_{\mathrm{size}}(x)$ is correct is $46.1\%$, which almost exactly matches the observed probability of $46.8\%$ that we obtain from training $\mathrm{GL}$-net to predict the dataset size in Table 4.

We further test our hypothesis empirically by training $\mathrm{GL}$-net on Imagenette class prediction (as in Table 3) and then summing up its class predictions to predict dataset size. On the Imagenette size-predictoin test set, $\mathrm{GL}$-net with this sum strategy achieves an accuracy of $43.5\pm 2.1\%$. This means $\mathrm{GL}$-net is likely learning a function with underlying mechanism similar to $f_{\mathrm{size}}$, where it predicts the presence of each class and then sums those predictions to output the total dataset size.

On the other hand, MLP + SVD correctly predicts class presence with probability $p=.656$. Predicting classes and summing up its individual predictions would give MLP + SVD a theoretical accuracy of $21\%$ by equation 38, which is significantly worse than its true performance of $73.1\%$. This suggests that MLP + SVD likely looks a characteristics other than class presence to determine dataset size.

## Appendix E LoRA Variants and Their Symmetries

In this section, we describe various LoRA variants that have been proposed for parameter-efficient finetuning. We also discuss their symmetries, and how one may process these LoRA variants with LoL models.

##### Training Modifications.

Several LoRA variants such as PiSSA (Meng et al., ) and LoRA+ (Hayou et al., ) have the same type of weight decomposition as the original LoRA, but with different initialization or training algorithms. These variants have the same exact symmetries as standard LoRA, so they can be processed in exactly the same way with our LoL models.

##### DoRA (Weight-Decomposed Low-Rank Adaptation).

Liu et al. () decompose the parameter updates into magnitude and directional components. For base weights $W\in\mathbb{R}^{n\times m}$, DoRA finetuning learns the standard low rank weights $U\in\mathbb{R}^{n\times r}$ and $V\in\mathbb{R}^{m\times r}$ along with a magnitude vector $\mathbf{m}\in\mathbb{R}^{m}$ so that the new finetuned weights are given by

$$ $(W+UV^{\top})\mathrm{Diag}\left(\frac{\mathbf{m}}{\left\lVert W+UV^{\top} \right\rVert_{c}}\right),$ (39) $$

where $\left\lVert\cdot\right\rVert_{c}:\mathbb{R}^{n\times m}\to\mathbb{R}^{m}$ is the norm of each column of the input matrix, the division $\frac{\mathbf{m}}{\left\lVert W+UV^{\top}\right\rVert_{c}}$ is taken elementwise, and $\mathrm{Diag}$ takes vectors in $\mathbb{R}^{m}$ to diagonal matrices in $\mathbb{R}^{m\times m}$. The vector $\mathbf{m}$ represents the norm of each column of the finetuned matrix. DoRA weights $(U,V,\mathbf{m})$ also have the same invertible matrix symmetry as standard LoRA, where $(UR,VR^{-\top},\mathbf{m})$ is functionally equivalent to $(U,V,\mathbf{m})$. The $\mathbf{m}$ vector cannot in general be changed without affecting the function. Thus, an LoL model could take as input $\mathbf{m}$ as an invariant feature, for instance by concatenating it to features in an invariant head of $\mathrm{GL}$-net, or concatenating it to the input of an MLP in the other LoL models.

##### KronA (Kronecker Adapter).

Edalati et al. () use a Kronecker product structure to decompose the learned weight matrix in a parameter-efficient way. For a weight matrix $W$ of shape $n\times m$, LoKr learns $U\in\mathbb{R}^{n^{\prime}\times m^{\prime}},V\in\mathbb{R}^{n^{\prime\prime}
\times m^{\prime\prime}}$ such that the finetuned weight is given by $W+U\otimes V$. Here, we no longer have a large general linear symmetry group. Instead, there is a scale symmetry, where $(U,V)$ is equivalent to $(sU,\frac{1}{s}V)$ for a scalar $s\in\mathbb{R}\setminus\{0\}$, since $(sU)\otimes(\frac{1}{s}V)=U\otimes V$. For LoL tasks, we can use a scale-invariant architecture, as for instance explored by Kalogeropoulos et al. (). We can also use $\mathrm{GL}$-net on the flattened $\mathrm{vec}(U)\in\mathbb{R}^{n^{\prime}m^{\prime}\times 1}$ and $\mathrm{vec}(V)\in\mathbb{R}^{n^{\prime\prime}m^{\prime\prime}\times 1}$, since $\mathbb{R}\setminus\{0\}=\mathrm{GL}(1)$ is the general linear symmetry group in the special case of rank $r=1$.

## Peer Reviews

confidence: 3
contribution: 4
initial_rating: 8
presentation: 3
questions: I have no major concerns and consider the paper as a strong submission, but it would be good if the authors could elaborate on the following issues:

1 - Can the proposed methodologies (such as GL-equivariant architectures) be directly applied to other LoRA-based fine-tuning variants, like DoRA (Liu et al., 2024)? If modifications are necessary for such variants, what aspects of the approach would need adaptation, particularly in handling their unique low-rank structures?


2 - Is the performance boost primarily due to the model’s increased number of parameters or computational requirements compared to other LoL techniques (Section 5.2, Table 2)? If so, how would you describe the cost/performance trade-off for GL-net in practical applications, especially in contexts where computational efficiency is prioritized?

3 - In Table 4, GL-net shows relatively lower performance on the dataset size prediction task. Could this lower performance be attributed to overfitting? If so, would any regularization techniques be effective here, and are there any architectural adjustments that could improve generalization on this specific task?


4 - The authors briefly mention future applications of GL-equivariant networks, such as model merging and LoRA editing (Conclusion)​. Could authors please elaborate on the specific challenges or potential approaches you foresee for these applications? In particular, how might the GL-net architecture be adjusted or extended for these tasks, which may have different requirements compared to the predictive tasks presented?


References

 1- Maron, Haggai, et al. "Invariant and equivariant graph networks." arXiv preprint arXiv:1812.09902 (2018).

 2 - Navon, Aviv, et al. "Equivariant architectures for learning in deep weight spaces." International Conference on Machine Learning. PMLR, 2023.

3- Liu, Shih-Yang, et al. "Dora: Weight-decomposed low-rank adaptation." arXiv preprint arXiv:2402.09353 (2024).
soundness: 3
strengths: 1 - The paper contributes to the field by capitalizing on the structural properties of LoRA weights, enabling efficient analysis and predictions without needing full model access or extensive retraining. By focusing on parameter symmetries in LoRA weights, the method facilitates tasks like predicting model performance on downstream tasks and analyzing fine-tuning attributes directly from the LoRA weights themselves without requiring compute- and parameter-heavy inferences. 

2 - The mathematical rigor in defining GL-invariant and GL-equivariant properties strengthens the reliability of the model design. In particular, Theorem 1 and Theorem 2 (Section 4)​ define the necessary properties of the proposed models, providing a clear foundation for future work in this domain. By establishing these invariance properties, the authors ensure that their architectures are theoretically grounded in symmetry-aware learning. The use of canonicalization and alignment methods for symmetry processing is well-supported by previous studies in geometric learning (see Maron et al., 2019) and further extends the concepts to the LoRA domain. 

3 - The paper demonstrates extensive experiments across multiple datasets (Section 5, Tables 2 and 3)​. Notably, the authors generate three novel datasets of LoRAs (Section 5.1)​, including CelebA-LoRA, Imagenette-LoRA, and Qwen2-ARC-LoRA, providing a robust basis for evaluating their models.
summary: This paper introduces Learning on LoRAs (LoL), an approach where low-rank adaptation weights, commonly used for efficient fine-tuning of large foundation models, serve as inputs for predictive models. The authors develop several GL(r)-equivariant and invariant neural architectures that process LoRA weights for various downstream tasks, such as predicting model accuracy, analyzing finetuning attributes, and performing data membership inference. The models leverage geometric deep learning techniques like canonicalization and invariant featurization, allowing efficient performance prediction without loss of structural information. The authors validate their approach through experimentation across diverse tasks on newly generated datasets of LoRA weights, including diffusion finetunes and language models finetunes.
weaknesses: 1 - The paper could benefit from a more thorough introduction to concepts like GL-invariance, O-invariance, and O-Align before discussing them within the context of Learning on LoRAs (LoL). While these concepts are central to the proposed architecture, they may be unfamiliar to some readers, especially those not specialized in geometric deep learning. Including a dedicated background section or a more extensive explanation would make the work more accessible and improve readability.

2 - GL-net’s superior performance may be partly due to a higher number of parameters or computational requirements compared to other LoL architectures. This could imply a cost-performance trade-off, where GL-net’s success on specific tasks may not be as scalable or efficient in scenarios requiring high throughput or limited compute resources. An analysis of this cost-performance trade-off would provide helpful insight into its practical applicability.

3 - The visuals, such as Figure 3 showcasing CLIP predictions, could benefit from clearer labeling and expanded captions to help interpret performance differences among models​. While the data is comprehensive, more explanatory notes within the figures would enhance accessibility for readers.

4 - The GL(r)-equivariant models, while theoretically sound, are complex and may pose practical challenges for adoption. The multi-step processing involving canonicalization, invariant featurization, and the invariant head layer adds architectural overhead (Section 3.3.3). While this complexity is justified for theoretical exploration, a discussion on trade-offs and simplifications would make it easier for practitioners to implement the approach. Previous work in weight-space learning, such as Navon et al. (2023), has shown simpler approaches that could be integrated as baselines to contrast model complexity and efficiency​.

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: 1. The LoRA model datasets used in this work are derived from Stable Diffusion V1.4 and Qwen2. A key question is whether the LoL model will remain effective if LoRA models are obtained from different versions, such as Stable Diffusion V2.1 or Qwen V2.5. 

2. Is there a clear rationale behind the choice of model architecture for the LoL models? Specifically, is using an MLP sufficient for the tasks at hand, or could other architectures be more effective?

3. The paper mentions that a large number of LoRA models were collected. How many GPU hours during the collection of these LoRA models?

4. The practical usage of the LoL models is not explicitly addressed. Are there any potential real-world applications except the dataset size prediction?
soundness: 2
strengths: 1. Several GL(r) equivariant and invariant neural architectures are proposed to effectively process weights of LoRAs.
2. Creating novel datasets for Learning on LoRAs
3. Authors conduct experiments across various finetuned models.
summary: In this paper, the authors explore the potential of Learning on LoRAs (LoL), a paradigm where LoRA weights are used as inputs to machine learning models. They fine-tune thousands of text-to-image diffusion models and language models to collect datasets of LoRAs and develop several symmetry-aware, invariant or equivariant LoL models using techniques such as canonicalization, invariant featurization, and equivariant layers.
weaknesses: 1. The detailed information regarding the models, such as the number of parameters and the training costs associated with different model structures and methods, is not sufficiently clear.
2. The motivation behind the task selection for LoL models is not clearly articulated. For instance, the LoL models aim to predict hyperparameters, CLIP scores, and training images of fine-tuned diffusion models using their LoRA weights. However, it is not immediately clear why these tasks are important or how they contribute to advancing the field. 
3. Other methods used for comparison in this work are not clearly defined and lacks sufficient discussion.

---

confidence: 3
contribution: 1
initial_rating: 3
presentation: 2
questions: - The model allows you to predict the accuracy of the model if a set of LoRA weights is applied; how is this useful? Why not just apply the LoRA weights and actually measure the accuracy?

- Why is it useful to predict how many data points the model was fine tuned on... If I have a set of LoRA adapters, I'll try every single one and the set of LoRA weights that does best is the one I'll use (regardless of how many data points it was trained on...).
soundness: 2
strengths: Shows that LoRA adapters have information in them that can be learned.
I guess the idea of learning a model for LoRA adapters is technically novel, then again the fact that weights encode information and that information can be extracted isn't a wild thought.
summary: The work introduces the Learning on LoRAs (LoL) paradigm; LoL is an ML model which has LoRA adapters as input and it outputs predictions about the weights.
weaknesses: I don't really understand why this is at all useful...

Meta-review: **Summary**


The paper introduces Learning on LoRAs (LoL), which performs regression/classification tasks on the low-rank adaptation parameters of a base model. To enable parameteric learning on the LoRA weights, the authors consider the inherent symmetries of these weights and propose various GL(r)-equivariant/invariant models to perform learning on these weights. Such parameteric model could be used for various downstream tasks, such as predicting model accuracy, analyzing finetuning attributes, and performing data membership inference. The authors validate their approach through experimentation across diverse tasks on newly generated datasets of LoRA weights, including LoRA adaptation paramters of diffusion and language models.


**Strengths**


* The paper proposes a novel problem, that could become of interest to the community. 
* The authors generate three novel datasets of LoRAs​, including CelebA-LoRA, Imagenette-LoRA, and Qwen2-ARC-LoRA, providing a robust basis for evaluating their models. If publicly released, such dataset would be of interest to the community.
* The careful treatment of the required invariances for designing the LoL models and the proposed different options are interesting. 


**Weaknesses**
 
* The paper does not provide clear details about the models, including the number of parameters and the training costs associated with various model structures and methods.
* While the proposed problem is interesting from a research perspective, its practical implications are not well motivated. In particular, LoL models aim to predict hyperparameters, CLIP scores, and training images of fine-tuned diffusion models using their LoRA weights. However, it is not immediately clear why these tasks are important or how they contribute to advancing the field. 
* It is not clear, whether the propose method can generalize across different base models (e.g., same architecture with different  parameters).


**Conclusion** 

he paper received polarized reviews. Reviewer EBmP offered a low-quality and dismissive critique, rating it 3 with a confidence of 3. Reviewer d8i6 was more positive, giving it an 8, though also with low confidence (confidence 3). Reviewer snsJ provided a more robust and confident review (rating 5, confidence 4), highlighting significant issues. The authors countered with a substantial rebuttal that addressed some concerns regarding generalization across base models. After thoroughly reviewing the paper, the other reviews, and the authors' rebuttal, my assessment aligns with Reviewer snsJ, finding the paper just below the acceptance threshold. The topic is compelling, but the practical implications are not convincingly presented, a concern also noted by Reviewer EBmP, despite the overall poor quality of their review. The inclusion of more concrete practical applications, such as using the method to detect if an adapted model is compromised (e.g., the training data contained backdoors), would significantly strengthen the paper. Given these considerations, I vote to reject this paper.

Decision: Reject