A Minimax Lower Bound for Low-Rank Matrix-Variate Logistic Regression

Logistic Regression (LR) is a statistical model used commonly in machine learning for classification problems [1]. In the simplest terms, LR seeks to model the conditional probability that a categorical response variable $y_{i}$ takes on one of two possible values, $y_{i}\in\{0,1\}$, which represents one of two possible events taking place (such as success/failure and detection/no detection). In regression analysis, the aim is to accurately estimate the model class parameters through a set of training data points $\{\mathbf{x}_{i},y_{i}\}_{i=1}^{n}$ , where $\mathbf{x}_{i}$ is the $i^{th}$ covariate sample vector. The conventional LR model is as follows:

where $y_{i}\in\{0,1\}$ is the binary response, $\mathbf{x}_{i}\in\mathbb{R}^{m}$ is the known covariate vector, $\mathbf{b}\in\mathbb{R}^{m}$ is the deterministic but unknown coefficient vector, and $z$ is the intercept where $\mathbb{E}[z]=0$.

In many practical applications, covariates naturally take the form of two-dimensional arrays. Common examples include images, biological data such as electroencephalography (EEG), fiber bundle imaging [2] and spatio-temporal data [3]. Classical machine learning techniques vectorize such matrix covariates and estimate a coefficient vector. This leads to computational inefficiency and the destruction of the underlying spatial structure of the coefficient matrix, which often carries valuable information [4, 5]. To address these issues, one can model the matrix LR problem as

where $\mathbf{X}_{i}\in\mathbb{R}^{m_{1}\times m_{2}}$ is the known covariate matrix, $\mathbf{B}\in\mathbb{R}^{m_{1}\times m_{2}}$ is the coefficient matrix, and $z$ is the intercept. The model in (2) preserves the matrix structure of, and the valuable information in, the covariate data. In matrix-variate logistic regression analysis, the goal is to find an estimate of the coefficient $\mathbf{B}$.

In this paper, we focus on the high-dimensional setting where $n\ll m_{1}m_{2}$ and derive a lower bound on the minimax risk of the matrix LR problem under the assumption that $\mathbf{B}$ has rank $r\ll\min\{m_{1},m_{2}\}$. We note that the low-rank assumption has been used ubiquitously in regression analysis in former works [6, 7]. This low-rank structure may arise from the presence of latent variables, such that the model’s intrinsic degrees of freedom are smaller than its extrinsic dimensionality. This allows us to represent the data in a lower-dimensional space and potentially reduce the sample complexity of estimating the parameters. Minimax lower bounds are useful in understanding the fundamental error thresholds of the problem under study, and in assessing the performance of the existing algorithms. They also provide beneficial insights to the parameters on which an achievable minimax risk might depend.

Whilst a number of works in the literature derive minimax lower bounds for higher-order linear regression problems [8, 7], as well as vector LR problems [6, 9, 10], to the best of our knowledge, little work has been done on this topic in matrix-variate LR problems. Regarding the existing literature, there are several works that study the matrix LR problem. Many works extend the matrix LR problem of (2) by proposing regularized matrix LR models in order to obtain rank-optimized or sparse estimates of the coefficient matrix in the model in (2) [5, 11]. Recently, works have introduced regularized matrix LR models for inference on image data [12]. Much of the literature develops efficient algorithms and provides empirical results on their performance [5, 11, 12] while some provide theoretical guarantees for the proposed algorithms [11]. By contrast, Baldin and Berthet [13] model the problem of graph regression as a matrix LR problem, where $\mathbf{B}\in\mathbb{R}^{m\times m}$. The matrix $\mathbf{B}$ is assumed to be block-sparse, low-rank, and square. Under these assumptions (which are restrictive and not the most general), the authors derive minimax lower bounds on the estimation error $\left\|\widehat{\mathbf{B}}-\mathbf{B}\right\|_{F}^{2}$.

The three prior works [4, 11, 12] implement algorithms for matrix logistic regression but do not prove sample complexity bounds (upper or lower). In this paper, we derive a minimax lower bound on the error of a low-rank LR model which gives a bound on the number of samples necessary for estimating $\mathbf{B}$. Contrary to prior works, we impose minimal assumptions on $\mathbf{B}$, making our results generalizable to a larger class of matrices.

Our Minimax lower bound relies on the distribution of the covariates and the energy of the regression matrix. Following previous works that derive minimax lower bounds in the dictionary learning setting [14, 15], we use the standard strategy of lower bounding the minimax risk in parametric estimation problems by the maximum error probability in a multiple hypothesis test problem and using Fano’s inequality [16]. We derive a lower bound that is proportional to the degrees of freedom in the rank-$r$ matrix LR problem, and we reduce the sample complexity from $\mathcal{O}(m_{1}m_{2})$ in the vector setting to $\mathcal{O}(r(m_{1}+m_{2}))$. This result is intuitively pleasing as it coincides with the number of free parameters in the model. We also show that our methods are easily extendable to the tensor case, i.e., $\underline{\mathbf{B}}$ is a coefficient tensor with some known properties.

We use the following notation convention throughout the paper: $x$, $\mathbf{x}$, $\mathbf{X}$ and $\underline{\mathbf{X}}$ denote scalars, vectors, matrices and tensors, respectively. We denote $\mathbf{I}_{m}$ as the $m\times m$ identity matrix. Also, $\left\|\mathbf{x}\right\|_{0}$, $\left\|\mathbf{x}\right\|_{2}$ and $\left\|\mathbf{X}\right\|_{F}$ denote the $\ell_{0}$, $\ell_{2}$ and Frobenius norms, respectively. Given a matrix $\mathbf{X}$, $\mathbf{x}_{j}$ is the $j^{th}$ column of $\mathbf{X}$. If $\underline{\mathbf{X}}$ is a third-order tensor, then by fixing indices in the first and second modes, the matrix $\underline{\mathbf{X}}_{:,:,j}$ is the $j^{th}$ frontal slice of $\underline{\mathbf{X}}$. If $a\in\mathbb{R}$, then $\lfloor a\rfloor$ is the greatest integer less than $a$. We denote $\mathbf{A}\otimes\mathbf{C}$ as the Kronecker product of matrices $\mathbf{A}$ and $\mathbf{C}$. We define the function $\mathrm{vec}(\mathbf{X})$ as the column-wise vectorization of matrix $\mathbf{X}$. Finally, $[R]$ is shorthand for $\{1,\dots,R\}$.

Consider the matrix LR model in (2), in which the Bernoulli response $y_{i}\in\{0,1\}$ is the $i^{th}$ response variable of $n$ independent and identically distributed (i.i.d) observations. The covariate matrix $\mathbf{X}_{i}\in\mathbb{R}^{m_{1}\times m_{2}}$ has independent, normally distributed, zero-mean and $\sigma^{2}$ variance entries. The response $y_{i}$ is generated according to (2) from $\mathbf{X}_{i}$ and a fixed coefficient matrix $\mathbf{B}\in\mathbb{R}^{m_{1}\times m_{2}}$ with $\left\|\mathbf{B}\right\|_{F}^{2}$ upper-bounded by a constant.

We consider the case where $\mathbf{B}$ is a rank-$r$ matrix. Specifically, the rank-$r$ singular value decomposition of $\mathbf{B}$ is

where $\mathbf{B}_{1}\in\mathbb{R}^{m_{1}\times r}$ and $\mathbf{B}_{2}\in\mathbb{R}^{m_{2}\times r}$ are (tall) singular vector matrices with orthonormal columns, and $\mathbf{G}=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{r})\in\mathbb{R}^{r\times r}$ is the matrix of singular values with $\lambda_{i}>0,\forall i\in[r]$. Under this low-rank structure of $\mathbf{B}$, we have

We use Kronecker product properties [17] to express $\mathbf{b}=\mathrm{vec}(\mathbf{B})$ as $\mathbf{b}=\left(\mathbf{B}_{2}\otimes\mathbf{B}_{1}\right)\mathrm{vec}(\mathbf{G})$.

The goal in LR is to find an estimate $\widehat{\mathbf{B}}$ of $\mathbf{B}$ using the training data $\bigl{\{}\mathbf{X}_{i},y_{i}\bigr{\}}_{i=1}^{n}$. We assume that the true coefficient matrix $\mathbf{B}$ belongs to the set

the open ball of radius $d$ with distance metric $\rho=\left\|\cdot\right\|_{F}$, which resides in a given the parameter space,

Thus $\mathcal{B}_{d}(\mathbf{0})\subset\mathcal{P}_{r}$ and $\mathbf{B}$ has energy bounded by $\left\|\mathbf{B}\right\|_{F}^{2}<d^{2}$.

Our objective is to find a lower bound on the minimax risk for estimating coefficient matrix $\mathbf{B}$. We define the non-decreasing function $\phi=\left\|\cdot\right\|_{F}^{2}\triangleq\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ with $\phi(0)=0$. The minimax risk is thus defined as the worst-case mean squared error (MSE) for the best estimator, i.e.,

where $\underline{\mathbf{X}}^{c}$ is the covariate tensor of $n$ samples. We point out that minimax risk in (7) is an inherent property of the LR problem and holds for all possible estimators.

The minimax lower bounds in the low-rank matrix LR setting that are based on Fano’s inequality will depend explicitly on the dimensions of the covariate matrices $\mathbf{X}_{i}$ and their distribution, the rank, the upper bound on the energy of the coefficient matrix $\mathbf{B}$, the number of samples, $n$, and the construction of the multiple hypothesis test set.

The novelty of our work is that we explicitly leverage the rank-$r$ structure of the coefficient matrix $\mathbf{B}$, in (3), which leads to the construction of each hypothesis that is structurally different in our setting compared to vector LR [6, 10, 9]. Furthermore, existing low-rank matrix packings, such as those in Negahban and Wainwright [18] are not useful in the LR setting of this paper, since the logistic function in our model makes part of our analysis non-trivial and fundamentally different to such works. In this work, we create a set $\mathcal{B}_{L}$ from three constructed sets (for $\mathbf{B}_{1}$, $\mathbf{B}_{2}$ and $\mathbf{G}$), and derive conditions under which all sets can exist. These conditions ensure the existence of a hypothesis set, $\mathcal{B}_{L}$, with elements that have a rank-$r$ matrix structure and additional essential properties noted below. We also show that our chosen constructed hypothesis set and analysis are more suitable for our matrix-variate model because they are easily generalizable to the tensor setting for LR.

The formal proof for Theorem 1 relies on three lemmas. Lemma 1 introduces the exponentially large, with respect to the dimensions, set of vectors from which we will later construct $\mathbf{G}$. The set is constructed as a subset of an $(r-1)$-dimensional hypercube with a required minimum distance between any two distinct elements in the set. Lemma 1 bounds the probability that this minimum distance is violated.

The above is a direct result of a standard application of McDiarmid’s inequality [21]. Notice, however, that our technique differs from [22] because we are imposing minimum distance conditions in the Hamming metric, rather than conditions on the inner product, between vectors.

Similar to the above, the following corollary introduces a set of matrices from which we will construct $\mathbf{B}_{1},\mathbf{B}_{2}$.

From the results in Lemma 1 and Corollary 1, Lemma 2 derives conditions on $L$ such that $\mathcal{B}_{L}$ with a certain set of properties exists, and constructs two sets of matrices: 1) $m_{1}\times r$ and $m_{2}\times r$ orthonormal matrices from the set of “generating” matrices defined in Corollary 1, and 2) $r\times r$ diagonal matrices from the set of “generating” vectors in Lemma 1. Each element $\mathbf{B}_{l}\in\mathcal{B}_{L}$ is constructed from the three sets defined above, according to the decomposition in (3). Next, we prove lower and upper bounds on the distance between any two distinct elements in $\mathcal{B}_{L}$. The lower bound determines the minimum packing distance between any two $\mathbf{B}_{l},\mathbf{B}_{l^{\prime}}$, whilst the upper bound is used to derive the results in Lemma 3.

Lemma 3 derives an upper bound on $\mathbb{I}(\mathbf{y};l|\underline{\mathbf{X}}^{c})$.

The final lemma bounds the error probability $\mathbb{P}(\widehat{l}(\mathbf{y})\neq l)$ under the conditions, mentioned in Section III-B, on $\varepsilon^{*}$ and $\left\|\widehat{\mathbf{B}}-\mathbf{B}_{l}\right\|_{F}^{2}$ for the recovery of hypothesis $\mathbf{B}_{l}$ by the minimum distance decoder. The proof follows exactly that for Lemma 8 in [14].

In this paper we provided a minimax lower bound on the low-rank matrix-variate LR problem in the high-dimensional setting. We constructed a packing set of low-rank structured matrices with finite energy. Using the construction, we derived bounds on the conditional mutual information defined in our problem, in order to obtain a lower bound on the minimax risk. Compared to the vector case, such as in [6], the result in Theorem 1 shows a decrease in the lower bound from $\mathcal{O}(m_{1}m_{2})$ to $\mathcal{O}(r(m_{1}+m_{2})$). The result also shows that the lower bound on the minimax risk is proportional to the intrinsic degrees of freedom in the coefficient matrix LR, (i.e., $r(m_{1}+m_{2}+1)$), and decreases with increasing sample size $n$. This suggests that we can develop algorithms that take advantage of the low-rank structure of the coefficient matrices. Moreover, the result in (9) can be generalized from low-rank matrices to low-rank tensors. Imposing a rank-$r$ decomposition on a coefficient tensor, $\underline{\mathbf{B}}$, holds the same conveniences as those discussed above of low-rank matrices. This low-rank structure on $\underline{\mathbf{B}}$ is a special case of the well-known Canonical Polyadic Decomposition or Parallel Factors (CANDECOMP/PARAFAC, or CP)[17], formally defined as,

where $\mathbf{b}_{k}^{(h)}\in\mathbb{R}^{m_{k}}$ is a column vector along the $k^{th}$ mode of $\underline{\mathbf{B}}$, $\circ$ is the outer product, and $\lambda_{h}\mathbf{b}_{1}^{(h)}\circ\cdots\circ\mathbf{b}_{K}^{(h)}$, for any rank $h$, is a rank-1 tensor weighted by $\lambda_{h}$. The rank-$r$ CP-decomposition expresses tensors as a sum of $r$ rank-1 tensors. Equivalent to (23) is the following expression of a CP-structured tensor:

where the tensor $\underline{\mathbf{G}}\in\mathbb{R}^{\overbrace{r\times\cdots\times r}^{K\text{-times}}}$ is simply a higher-order analogue of the diagonal matrix $\mathbf{G}$ in (3) (where only the elements along the super-diagonal of $\underline{\mathbf{G}}$ are non-zero), and the mode-$k$ factor matrices $\mathbf{B}_{k}\in\mathbb{R}^{m_{k}\times r},~{}\forall k\in[K]$ are rank-$r$ matrices with orthonormal columns. Thus, the setup and construction proposed in this paper can be extended to the tensor case, where $\mathbf{B}$ is simply a special case of $\underline{\mathbf{B}}$, with $K=2$ factor matrices.