Product Kernels for Joint Similarity

Why Multiply Omics Kernels?

Two patients can look similar in one omics view and quite different in another. Additive kernel fusion can retain a strong signal from one view even when the other views disagree. A product kernel encodes a different goal: it gives a high joint similarity primarily to pairs that are similar across all included views. This can be useful when the scientific notion of resemblance requires agreement among, for example, transcriptomic, proteomic, and metabolomic measurements.

The product is taken entry by entry, not by ordinary matrix multiplication. Its interpretation depends on the input kernels: in particular, kernels with nonnegative entries support a direct multiplicative reading of pairwise similarity. Positive semidefiniteness alone does not guarantee that every entry is nonnegative.

Definition: Suppose $n$ patients have been measured in $V$ omics views, and $K^{(v)}$ is the $n\times n$ kernel matrix for view $v$. The elementwise product kernel $K^{(\mathrm{prod})}$ is defined by $$ K^{(\mathrm{prod})}_{ij}=\prod_{v=1}^{V}K^{(v)}_{ij},\qquad i,j=1,\ldots,n. $$ Equivalently, $K^{(\mathrm{prod})}=K^{(1)}\circ\cdots\circ K^{(V)}$, where $\circ$ denotes the Hadamard (elementwise) product.

For this entrywise operation to describe the same patient pairs across views, every input matrix must use the same patients in the same row and column order. For the standard kernel guarantee, each $K^{(v)}$ must be symmetric and positive semidefinite (PSD): for every vector $c\in\mathbb{R}^n$, $c^\mathsf{T}K^{(v)}c\geq 0$. A kernel matrix may also have nonnegative entries, which is an additional property useful for interpreting multiplication as joint similarity.

Why the Product Is Still a Kernel

A product of PSD kernels is PSD, a result known as the Schur product theorem for matrices. The reason can be seen from the feature-map representation of a kernel. For each view $v$, write $K^{(v)}_{ij}=\langle\phi_v(i),\phi_v(j)\rangle$, where $\phi_v(i)$ is the feature representation of patient $i$ in that view. Define a joint feature representation by taking the tensor product of the view-specific representations:

\psi(i)=\phi_1(i)\otimes\cdots\otimes\phi_V(i).

The inner product of two tensor products is the product of their component inner products. Thus the product kernel is exactly the Gram matrix of the joint feature representations:

\langle\psi(i),\psi(j)\rangle =\prod_{v=1}^{V}\langle\phi_v(i),\phi_v(j)\rangle =\prod_{v=1}^{V}K^{(v)}_{ij} =K^{(\mathrm{prod})}_{ij}.

To check positive semidefiniteness directly, take any $c\in\mathbb{R}^n$. Then

c^\mathsf{T}K^{(\mathrm{prod})}c =\sum_{i=1}^{n}\sum_{j=1}^{n}c_i c_j \langle\psi(i),\psi(j)\rangle =\left\|\sum_{i=1}^{n}c_i\psi(i)\right\|^2 \geq 0.

This argument establishes the guarantee when the input matrices are Gram matrices of valid kernels on the same patients. It does not establish that product entries are probabilities, nor does it ensure nonnegative entries unless the inputs have them.

Worked Example: Multiplying Two Views Entry by Entry

Suppose three hypothetical patients, A, B, and C, have two omics kernel matrices

K^{(1)}= \begin{pmatrix} 1&0.80&0.30\\ 0.80&1&0.30\\ 0.30&0.30&1 \end{pmatrix}, \qquad K^{(2)}= \begin{pmatrix} 1&0.20&0.70\\ 0.20&1&0.20\\ 0.70&0.20&1 \end{pmatrix}.

Assume these are hypothetical PSD kernel matrices, with patients in the same order in both views. Their entries are nonnegative, so the pairwise products can be read as joint similarities. For A and B, the similarity is $0.80$ in view 1 but only $0.20$ in view 2. For A and C, the values are $0.30$ and $0.70$. The product matrix is

K^{(\mathrm{prod})}= \begin{pmatrix} 1&0.80(0.20)&0.30(0.70)\\ 0.80(0.20)&1&0.30(0.20)\\ 0.30(0.70)&0.30(0.20)&1 \end{pmatrix} = \begin{pmatrix} 1&0.160&0.210\\ 0.160&1&0.060\\ 0.210&0.060&1 \end{pmatrix}.

The arithmetic shows the joint effect: A and B receive similarity $0.160$, while A and C receive $0.210$, because A and C have more balanced similarities across these two views. The diagonals remain 1 because each input diagonal is 1. By the product-kernel result, the product is PSD. The stated interpretation assumes the displayed input matrices are valid PSD kernels and that their patient ordering agrees.

What the Product Emphasizes

When all input entries are between 0 and 1, multiplication attenuates a pair's similarity whenever any view assigns it a low value. If one view assigns a pair a similarity of zero, its product similarity is zero regardless of the other views. This is the sense in which the product emphasizes agreement: a pair must receive support across views to retain a large product. It is not a formal test of agreement, and a large product does not establish a biological mechanism.

The multiplication also makes scale important. A kernel whose typical entries are much smaller than those of the other kernels can shrink many product entries. Normalizing a kernel's scale may therefore matter for interpretation; diagonal normalization is one common choice when nonzero diagonals allow it. Such preprocessing changes the input matrices and should be described, not silently assumed. With many views, repeated multiplication can make off-diagonal values very small, even if each individual view reports moderate similarity.

Worked Example: Comparing Pairs Using Gaussian Kernels

Consider two hypothetical patients in two omics views. Suppose each view uses the Gaussian kernel $K_{ij}=\exp(-d_{ij}^{2})$, with the same parameter in both views, where $d_{ij}$ is the distance between the patients in that view. The Gaussian kernel is PSD for Euclidean feature vectors, and its entries are positive. For pair A, distances are $0.5$ and $1.5$; for pair B, distances are $0.8$ and $0.8$.

For pair A, the product is

\exp(-0.5^2)\exp(-1.5^2) =\exp(-0.25-2.25) =\exp(-2.5) \approx 0.08208.

For pair B, it is

\exp(-0.8^2)\exp(-0.8^2) =\exp(-0.64-0.64) =\exp(-1.28) \approx 0.2780.

Pair A is very close in one view but distant in the other, whereas pair B has moderate distances in both. The product ranks B as more jointly similar: $0.2780>0.08208$. This conclusion follows from the stated distances and the common Gaussian-kernel form. If different Gaussian parameters or input scales were used across views, those choices would also influence the comparison.

Worked Example: A Weak View Can Dominate the Joint Similarity

Suppose three hypothetical views give a pair of patients similarities $0.80$, $0.75$, and $0.90$. A second pair has similarities $0.95$, $0.95$, and $0.20$. Assume each value is an entry from a PSD kernel with nonnegative values, and the patients are aligned across the three views. Their product similarities are

0.80(0.75)(0.90)=0.5400, \qquad 0.95(0.95)(0.20)=0.1805.

The second pair is highly similar in two views, but the low value in the third view reduces its joint similarity to $0.1805$. This is appropriate if similarity is intended to require evidence from every view. It may be undesirable if the third view is noisy or irrelevant. The arithmetic is valid for the stated inputs; the scientific decision to require all three views is a modeling choice, not a consequence of the PSD guarantee.

Assumptions, Diagnostics, and Alternatives

Before interpreting a product kernel, check the conditions that make the calculation meaningful and the resulting matrix suitable for downstream kernel methods.

Align patients.
Confirm that every matrix refers to the same patients in the same row and column order. A misordered matrix multiplies similarities for different pairs and makes the result uninterpretable.

Check each input kernel.
Confirm symmetry and positive semidefiniteness, using the kernel construction or an eigenvalue diagnostic. Small negative eigenvalues can occur from numerical error; substantial negative eigenvalues call the kernel assumption into question.

Inspect entries and scale.
Review the ranges, diagonals, and distributions of off-diagonal entries. If a direct joint-similarity interpretation is intended, check that entries are nonnegative. Look for one view with much smaller typical values or many near-zero values.

Inspect the product.
Check whether off-diagonal values have collapsed toward zero, whether some patients have almost no appreciable similarities, and whether the product still distinguishes pairs in a scientifically useful way.

A central failure mode is treating PSD and nonnegative entries as interchangeable. A PSD kernel can have negative entries, especially after centering. Its elementwise product with another PSD kernel remains PSD, but negative products or products of two negative entries do not have the simple interpretation “similar in every view.” Another pitfall is multiplying kernels from different patient sets or orders; the resulting array can still contain numbers, but they no longer represent corresponding pairs.

The additive kernel fusion discussed in the earlier material on additive kernel fusion is the main conceptual alternative. Addition can preserve a strong contribution from one view when other views are weak, which suits complementary signals. Multiplication instead penalizes disagreement and suits a joint-similarity objective. Neither operation is universally preferable: the choice depends on whether the scientific target is shared resemblance across views or useful signal contributed by any view. The product is also sensitive to the number of views and to view-specific scale, and it can suppress pairs because of noise in a single view. It does not automatically resolve missing measurements, select relevant views, or establish that the fused similarity captures the intended biology.

Key takeaway: An elementwise product of PSD kernels is PSD, and with nonnegative inputs its entries multiply view-specific similarities. That same multiplicative structure makes the result demanding: a weak value in any included view can strongly reduce a pair's joint similarity.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two aligned kernels give a patient pair similarities $0.60$ and $0.40$. What is the product similarity, and what does it mean under nonnegative-entry assumptions?
Why does an elementwise product of PSD kernels remain PSD?
A pair has values $0.90$, $0.85$, and $0.05$ across three views. Explain why its product may be small even though two values are high.
What is wrong with multiplying two kernel matrices that contain the same patients in different row orders?
Give one situation in which additive fusion may be preferable to product fusion.

Solutions. (1) $0.60(0.40)=0.240$; under nonnegative-entry assumptions, this is the pair's multiplicative joint similarity across the two views. (2) Each PSD kernel has a feature-map representation, and the product is the Gram matrix of tensor-product features, so every quadratic form is a squared norm and is nonnegative. (3) The product is $0.90(0.85)(0.05)=0.03825$; the third view strongly attenuates it. (4) The same matrix indices would refer to different patient pairs, so entrywise multiplication would combine unrelated comparisons. (5) Additive fusion may be preferable when views contain complementary signals and a strong similarity in one view should not be erased by weak similarities in others.

Product Kernels for Joint Similarity

What You'll Learn